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1l5aifh9z | maths | matrices-and-determinants | symmetric-and-skew-symmetric-matrices | <p>Let $$A = \left[ {\matrix{
0 & { - 2} \cr
2 & 0 \cr
} } \right]$$. If M and N are two matrices given by $$M = \sum\limits_{k = 1}^{10} {{A^{2k}}} $$ and $$N = \sum\limits_{k = 1}^{10} {{A^{2k - 1}}} $$ then MN<sup>2</sup> is :</p> | [{"identifier": "A", "content": "a non-identity symmetric matrix"}, {"identifier": "B", "content": "a skew-symmetric matrix"}, {"identifier": "C", "content": "neither symmetric nor skew-symmetric matrix"}, {"identifier": "D", "content": "an identity matrix"}] | ["A"] | null | <p>$$A = \left[ {\matrix{
0 & { - 2} \cr
2 & 0 \cr
} } \right]$$</p>
<p>$${A^2} = \left[ {\matrix{
0 & { - 2} \cr
2 & 0 \cr
} } \right]\left[ {\matrix{
0 & { - 2} \cr
2 & 0 \cr
} } \right] = \left[ {\matrix{
{ - 4} & 0 \cr
0 & { - 4} \cr
} } \right] = - 4I$$</p>
<p>$$M =... | mcq | jee-main-2022-online-25th-june-morning-shift |
1l6nl9px1 | maths | matrices-and-determinants | symmetric-and-skew-symmetric-matrices | <p>Let $$\mathrm{A}$$ and $$\mathrm{B}$$ be any two $$3 \times 3$$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?</p> | [{"identifier": "A", "content": "$$\\mathrm{A}^{4}-\\mathrm{B}^{4}$$ is a smmetric matrix"}, {"identifier": "B", "content": "$$\\mathrm{AB}-\\mathrm{BA}$$ is a symmetric matrix"}, {"identifier": "C", "content": "$$\\mathrm{B}^{5}-\\mathrm{A}^{5}$$ is a skew-symmetric matrix"}, {"identifier": "D", "content": "$$\\mathrm... | ["C"] | null | <p>(A) $$M = {A^4} - {B^4}$$</p>
<p>$${M^T} = {({A^4} - {B^4})^T} = {({A^T})^4} - {({B^T})^4}$$</p>
<p>$$ = {A^4} - {( - B)^4} = {A^4} - {B^4} = M$$</p>
<p>(B) $$M = AB - BA$$</p>
<p>$${M^T} = {(AB - BA)^T} = {(AB)^T} - {(BA)^T}$$</p>
<p>$$ = {B^T}{A^T} - {A^T}{B^T}$$</p>
<p>$$ = - BA - A( - B)$$</p>
<p>$$ = AB - BA =... | mcq | jee-main-2022-online-28th-july-evening-shift |
1ldsgfp13 | maths | matrices-and-determinants | symmetric-and-skew-symmetric-matrices | <p>Let A be a symmetric matrix such that $$\mathrm{|A|=2}$$ and $$\left[ {\matrix{
2 & 1 \cr
3 & {{3 \over 2}} \cr
} } \right]A = \left[ {\matrix{
1 & 2 \cr
\alpha & \beta \cr
} } \right]$$. If the sum of the diagonal elements of A is $$s$$, then $$\frac{\beta s}{\alpha^2}$$ is... | [] | null | 5 | <p>$$A = \left( {\matrix{
a & c \cr
c & b \cr
} } \right)$$</p>
<p>$$|A| = ab - {c^2} = 2$$ ...... (1)</p>
<p>$$\left( {\matrix{
2 & 1 \cr
3 & {{3 \over 2}} \cr
} } \right)\left( {\matrix{
a & c \cr
c & b \cr
} } \right) = \left( {\matrix{
1 & 2 \cr
\alpha & \beta \cr
}... | integer | jee-main-2023-online-29th-january-evening-shift |
1ldu4mkm0 | maths | matrices-and-determinants | symmetric-and-skew-symmetric-matrices | <p>Let A, B, C be 3 $$\times$$ 3 matrices such that A is symmetric and B and C are skew-symmetric. Consider the statements</p>
<p>(S1) A$$^{13}$$ B$$^{26}$$ $$-$$ B$$^{26}$$ A$$^{13}$$ is symmetric</p>
<p>(S2) A$$^{26}$$ C$$^{13}$$ $$-$$ C$$^{13}$$ A$$^{26}$$ is symmetric</p>
<p>Then,</p> | [{"identifier": "A", "content": "Only S2 is true"}, {"identifier": "B", "content": "Only S1 is true"}, {"identifier": "C", "content": "Both S1 and S2 are false"}, {"identifier": "D", "content": "Both S1 and S2 are true"}] | ["A"] | null | $A^{T}=A, B^{T}=-B, C^{T}=-C$
<br/><br/>
$$
\begin{aligned}
P & =A^{13} B^{26}-B^{26} A^{13} \\\\
P^{T} & =\left(A^{13} B^{26}-B^{26} A^{13}\right)^{T}=\left(A^{13} B^{26}\right)^{T}-\left(B^{26} A^{B}\right)^{T} \\\\
& =\left(B^{26}\right)^{T}\left(A^{13}\right)^{T}-\left(A^{13}\right)^{T}\left(B^{26}\right)^{T} \\\\
... | mcq | jee-main-2023-online-25th-january-evening-shift |
1lgq0m4hk | maths | matrices-and-determinants | symmetric-and-skew-symmetric-matrices | <p>The number of symmetric matrices of order 3, with all the entries from the set $$\{0,1,2,3,4,5,6,7,8,9\}$$ is :</p> | [{"identifier": "A", "content": "$$10^{9}$$"}, {"identifier": "B", "content": "$$9^{10}$$"}, {"identifier": "C", "content": "$$10^{6}$$"}, {"identifier": "D", "content": "$$6^{10}$$"}] | ["C"] | null | <p>Sure! A symmetric matrix is a square matrix that is equal to its transpose. For a matrix to be symmetric, the element at row i and column j must be equal to the element at row j and column i. In other words, $$A_{ij} = A_{ji}$$. </p>
<p>For a 3 $$ \times $$ 3 symmetric matrix, it looks like this:</p>
<p>$$
\begin{pm... | mcq | jee-main-2023-online-13th-april-morning-shift |
adqLo6ekAzwOBx7J | maths | matrices-and-determinants | trace-of-a-matrix | Let $$A$$ be $$a\,2 \times 2$$ matrix with real entries. Let $$I$$ be the $$2 \times 2$$ identity matrix. Denote by tr$$(A)$$, the sum of diagonal entries of $$a$$. Assume that $${a^2} = I.$$
<br/><b>Statement-1 :</b> If $$A \ne I$$ and $$A \ne - I$$, then det$$(A)=-1$$
<br/><b>Statement- 2 :</b> If $$A \ne I$$ and $... | [{"identifier": "A", "content": "statement - 1 is false, statement -2 is true "}, {"identifier": "B", "content": "statement -1 is true, statement - 2 is true; statement - 2 is a correct explanation for statement - 1."}, {"identifier": "C", "content": "statement - 1 is true, statement - 2 is true; statement - 2 is not a... | ["D"] | null | Let $$A = \left[ {\matrix{
a & b \cr
c & d \cr
} } \right]$$ $$\,\,\,$$ then $${A^2} = 1$$
<br><br>$$ \Rightarrow {a^2} + bc = 1\,\,\,\,ab + bd = 0$$
<br><br>$$ac + cd = 0\,\,\,\,bc + {d^2} = 1$$
<br><br>From these four relations,
<br><br>$${a^2} + bc = bc + {d^2} \Rightarrow {a^2} = {d^2}$$
<br><b... | mcq | aieee-2008 |
hHBIhG3bWMTtRnpl | maths | matrices-and-determinants | trace-of-a-matrix | Let $$A$$ be a $$\,2 \times 2$$ matrix with non-zero entries and let $${A^2} = I,$$
<br/>where $$I$$ is $$2 \times 2$$ identity matrix. Define
<br/>$$Tr$$$$(A)=$$ sum of diagonal elements of $$A$$ and $$\left| A \right| = $$ determinant of matrix $$A$$.
<br/><b>Statement- 1:</b> $$Tr$$$$(A)=0$$.
<br/><b>Statement- 2:... | [{"identifier": "A", "content": "statement - 1 is true, statement - 2 is true; statement - 2 is <b>not</b> a correct explanation for statement - 1. "}, {"identifier": "B", "content": "statement - 1 is true, statement - 2 is false. "}, {"identifier": "C", "content": "statement - 1 is false, statement -2 is true "}, {"... | ["B"] | null | Let $$A = \left( {\matrix{
a & b \cr
c & d \cr
} } \right)$$ where $$a,b,c,d$$ $$ \ne 0$$
<br><br>$${A^2} = \left( {\matrix{
a & b \cr
c & d \cr
} } \right)\left( {\matrix{
a & b \cr
c & d \cr
} } \right)$$
<br><br>$$ \Rightarrow {A^2} = \left( {\matrix{
{... | mcq | aieee-2010 |
0siKIhXARnH2944i8v7k9k2k5gzyxdt | maths | matrices-and-determinants | trace-of-a-matrix | The number of all 3 × 3 matrices A, with
enteries from the set {–1, 0, 1} such that the sum
of the diagonal elements of AA<sup>T</sup> is 3, is | [] | null | 672 | Let A = $$\left[ {\matrix{
{{a_{11}}} & {{a_{12}}} & {{a_{13}}} \cr
{{a_{21}}} & {{a_{22}}} & {{a_{23}}} \cr
{{a_{31}}} & {{a_{32}}} & {{a_{33}}} \cr
} } \right]$$
<br><br>$$ \therefore $$ A<sup>T</sup> = $$\left[ {\matrix{
{{a_{11}}} & {{a_{21}}} & {{a_{31}}} \cr
... | integer | jee-main-2020-online-8th-january-morning-slot |
Awf3Xo25ZkDyIfzHoijgy2xukewq8sg6 | maths | matrices-and-determinants | trace-of-a-matrix | Let A be a 2 $$ \times $$ 2 real matrix with entries from
{0, 1} and |A|
$$ \ne $$ 0. Consider the following two
statements :
<br/><br/>(P) If A $$ \ne $$ I<sub>2</sub>
, then |A| = –1
<br/>(Q) If |A| = 1, then tr(A) = 2,
<br/><br/>where I<sub>2</sub>
denotes 2 $$ \times $$ 2 identity matrix and tr(A)
denotes the sum ... | [{"identifier": "A", "content": "(P) is true and (Q) is false"}, {"identifier": "B", "content": "Both (P) and (Q) are false"}, {"identifier": "C", "content": "Both (P) and (Q) are true"}, {"identifier": "D", "content": "(P) is false and (Q) is true"}] | ["D"] | null | Let A = $$\left[ {\matrix{
a & b \cr
c & d \cr
} } \right]$$, where a, b, c, d $$ \in $$ {0, 1}
<br><br>$$ \Rightarrow $$ |A| = ad – bc
<br><br>$$ \therefore $$ ad = 0 or 1 and bc = 0 or 1
<br><br>So possible values of |A| are 1, 0 or –1
<br><br>(P) If A $$ \ne $$ I<sub>2</sub>
then |A| is either ... | mcq | jee-main-2020-online-2nd-september-morning-slot |
YI4Yw7Wd7pYuECq7od1kmhzsuuy | maths | matrices-and-determinants | trace-of-a-matrix | The total number of 3 $$\times$$ 3 matrices A having entries from the set {0, 1, 2, 3} such that the sum of all the diagonal entries of AA<sup>T</sup> is 9, is equal to _____________. | [] | null | 766 | $$A{A^T} = \left[ {\matrix{
x & y & z \cr
a & b & c \cr
d & e & f \cr
} } \right]\left[ {\matrix{
x & a & d \cr
y & b & e \cr
z & c & f \cr
} } \right]$$<br><br>$$ = \left[ {\matrix{
{{x^2} + {y^2} + {z^2}} & {ax + by + cz} & {... | integer | jee-main-2021-online-16th-march-morning-shift |
p9s7EngSW3BQIbwlq71kmlj61ex | maths | matrices-and-determinants | trace-of-a-matrix | Let $$A + 2B = \left[ {\matrix{
1 & 2 & 0 \cr
6 & { - 3} & 3 \cr
{ - 5} & 3 & 1 \cr
} } \right]$$ and $$2A - B = \left[ {\matrix{
2 & { - 1} & 5 \cr
2 & { - 1} & 6 \cr
0 & 1 & 2 \cr
} } \right]$$. If Tr(A) denotes the sum of all diagonal ... | [{"identifier": "A", "content": "1"}, {"identifier": "B", "content": "2"}, {"identifier": "C", "content": "0"}, {"identifier": "D", "content": "3"}] | ["B"] | null | $$A = {1 \over 5}((A + 2B) + 2(2A - B))$$<br><br>$$ = {1 \over 5}\left( {\left[ {\matrix{
1 & 2 & 0 \cr
6 & { - 3} & 3 \cr
{ - 5} & 3 & 1 \cr
} } \right] + \left[ {\matrix{
4 & { - 2} & {10} \cr
4 & { - 2} & {12} \cr
0 & 2 & 4 \cr
} } \ri... | mcq | jee-main-2021-online-18th-march-morning-shift |
1ldpsteev | maths | matrices-and-determinants | trace-of-a-matrix | <p>Let $$A = \left( {\matrix{
1 & 0 & 0 \cr
0 & 4 & { - 1} \cr
0 & {12} & { - 3} \cr
} } \right)$$. Then the sum of the diagonal elements of the matrix $${(A + I)^{11}}$$ is equal to :</p> | [{"identifier": "A", "content": "4094"}, {"identifier": "B", "content": "2050"}, {"identifier": "C", "content": "6144"}, {"identifier": "D", "content": "4097"}] | ["D"] | null | $A^{2}=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3\end{array}\right]\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3\end{array}\right]$
<br/><br/>$$
=\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 4 & -1 \\
0 & 12 & -3
\end{array}\right]=A
$$
<br/><br/>$$\Rightarrow \mathrm{A}_{3}=\mathrm{A... | mcq | jee-main-2023-online-31st-january-morning-shift |
jaoe38c1lscom6c6 | maths | matrices-and-determinants | trace-of-a-matrix | <p>Let $$A$$ be a $$2 \times 2$$ real matrix and $$I$$ be the identity matrix of order 2. If the roots of the equation $$|\mathrm{A}-x \mathrm{I}|=0$$ be $$-1$$ and 3, then the sum of the diagonal elements of the matrix $$\mathrm{A}^2$$ is</p> | [] | null | 10 | <p>$$|A-x I|=0$$</p>
<p>Roots are $$-$$1 and 3</p>
<p>Sum of roots $$=\operatorname{tr}(A)=2$$</p>
<p>Product of roots $$=|\mathrm{A}|=-3$$</p>
<p>Let $$A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$$</p>
<p>We have $$\mathrm{a}+\mathrm{d}=2$$</p>
<p>$$\mathrm{ad}-\mathrm{bc}=-3$$</p>
<p>$$A^2=\left[\begin{... | integer | jee-main-2024-online-27th-january-evening-shift |
1lsg3zn8d | maths | matrices-and-determinants | trace-of-a-matrix | <p>Let $$R=\left(\begin{array}{ccc}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right)$$ be a non-zero $$3 \times 3$$ matrix, where $$x \sin \theta=y \sin \left(\theta+\frac{2 \pi}{3}\right)=z \sin \left(\theta+\frac{4 \pi}{3}\right) \neq 0, \theta \in(0,2 \pi)$$. For a square matrix $$M$$, le... | [{"identifier": "A", "content": "Only (I) is true\n"}, {"identifier": "B", "content": "Only (II) is true\n"}, {"identifier": "C", "content": "Both (I) and (II) are true\n"}, {"identifier": "D", "content": "Neither (I) nor (II) is true"}] | ["D"] | null | <p>$$\begin{aligned}
& x \sin \theta=y \sin \left(\theta+\frac{2 \pi}{3}\right)=z \sin \left(\theta+\frac{4 \pi}{3}\right) \neq 0 \\
& \Rightarrow x, y, z \neq 0
\end{aligned}$$</p>
<p>Also,</p>
<p>$$\begin{aligned}
& \sin \theta+\sin \left(\theta+\frac{2 \pi}{3}\right)+\sin \left(\theta+\frac{4 \pi}{3}\right)=0 \foral... | mcq | jee-main-2024-online-30th-january-evening-shift |
lv0vxdis | maths | matrices-and-determinants | trace-of-a-matrix | <p>Let $$A$$ be a square matrix of order 2 such that $$|A|=2$$ and the sum of its diagonal elements is $$-$$3 . If the points $$(x, y)$$ satisfying $$\mathrm{A}^2+x \mathrm{~A}+y \mathrm{I}=\mathrm{O}$$ lie on a hyperbola, whose transverse axis is parallel to the $$x$$-axis, eccentricity is $$\mathrm{e}$$ and the lengt... | [] | null | 25 | <p>$$|A|=2 \sum \mathrm{dia}=-3$$</p>
<p>$$\therefore \quad$$ character equation : $$A^2+3 A+2 I=0$$</p>
<p>$$\Rightarrow x=3 \quad y=2$$</p>
<p>$$\because$$ We are getting only one point $$(3,2)$$ but its given many points satisfy this equation.</p>
<p>Moreover hyperbola whose transverse axis is $$x$$ axis and passing... | integer | jee-main-2024-online-4th-april-morning-shift |
lv5grwjl | maths | matrices-and-determinants | trace-of-a-matrix | <p>Let $$A=\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]$$. If the sum of the diagonal elements of $$A^{13}$$ is $$3^n$$, then $$n$$ is equal to ________.</p> | [] | null | 7 | <p>$$\begin{aligned}
& A=\left[\begin{array}{cc}
2 & -1 \\
1 & 1
\end{array}\right] \\
& A^2=\left[\begin{array}{cc}
2 & -1 \\
1 & 1
\end{array}\right]\left[\begin{array}{cc}
2 & -1 \\
1 & 1
\end{array}\right] \\
& A^2=\left[\begin{array}{cc}
3 & -3 \\
3 & 0
\end{array}\right]=3\left[\begin{array}{cc}
1 & -1 \\
1 & 0
\... | integer | jee-main-2024-online-8th-april-morning-shift |
amsDpowk5keRB2xC | maths | matrices-and-determinants | transpose-of-a-matrix | If $$A = \left[ {\matrix{
1 & 2 & 2 \cr
2 & 1 & { - 2} \cr
a & 2 & b \cr
} } \right]$$ is a matrix satisfying the equation
<br/><br/>$$A{A^T} = 9\text{I},$$ where $$I$$ is $$3 \times 3$$ identity matrix, then the ordered
<br><br/>pair $$(a, b)$$ is equal to :</br> | [{"identifier": "A", "content": "$$(2, 1)$$"}, {"identifier": "B", "content": "$$(-2, -1)$$"}, {"identifier": "C", "content": "$$(2, -1)$$"}, {"identifier": "D", "content": "$$(-2, 1)$$"}] | ["B"] | null | $$\left[ {\matrix{
1 & 2 & 2 \cr
2 & 1 & { - 2} \cr
a & 2 & b \cr
} } \right]\left[ {\matrix{
1 & 2 & a \cr
2 & 1 & 2 \cr
2 & { - 2} & b \cr
} } \right] = \left[ {\matrix{
9 & 0 & 0 \cr
0 & 9 & 0 \cr
0 & 0... | mcq | jee-main-2015-offline |
qS95ij4pasrt0BwY2C0oD | maths | matrices-and-determinants | transpose-of-a-matrix | If P = $$\left[ {\matrix{
{{{\sqrt 3 } \over 2}} & {{1 \over 2}} \cr
{ - {1 \over 2}} & {{{\sqrt 3 } \over 2}} \cr
} } \right],A = \left[ {\matrix{
1 & 1 \cr
0 & 1 \cr
} } \right]\,\,\,$$
<br/><br/>Q = PAP<sup>T</sup>, then P<sup>T</sup> Q<sup>2015</sup> P is : | [{"identifier": "A", "content": "$$\\left[ {\\matrix{\n 0 & {2015} \\cr \n 0 & 0 \\cr \n\n } } \\right]$$"}, {"identifier": "B", "content": "$$\\left[ {\\matrix{\n {2015} & 1 \\cr \n 0 & {2015} \\cr \n\n } } \\right]$$ "}, {"identifier": "C", "content": "$$\\left[ {\\matrix{\n {2015} &... | ["D"] | null | P = $$\left[ {\matrix{
{{{\sqrt 3 } \over 2}} & {{1 \over 2}} \cr
{ - {1 \over 2}} & {{{\sqrt 3 } \over 2}} \cr
} } \right]$$
<br><br>$$ \therefore $$ P<sup>T</sup> = $$\left[ {\matrix{
{{{\sqrt 3 } \over 2}} & { - {1 \over 2}} \cr
{{1 \over 2}} & {{{\sqrt 3 } \over ... | mcq | jee-main-2016-online-9th-april-morning-slot |
2fwdocUS7iYG0xGqpGtLt | maths | matrices-and-determinants | transpose-of-a-matrix | For two 3 × 3 matrices A and B, let A + B = 2B<sup>T</sup> and 3A + 2B = I<sub>3</sub>, where B<sup>T</sup> is
the transpose of B and I<sub>3</sub> is 3 × 3 identity matrix. Then : | [{"identifier": "A", "content": "5A + 10B = 2I<sub>3</sub> "}, {"identifier": "B", "content": "10A + 5B = 3I<sub>3</sub>"}, {"identifier": "C", "content": "B + 2A = I<sub>3</sub>"}, {"identifier": "D", "content": "3A + 6B = 2I<sub>3</sub>"}] | ["B"] | null | Given, A + B = 2B<sup>T</sup> .......(1)
<br><br>$$ \Rightarrow $$ (A + B)<sup>T</sup> = (2B<sup>T</sup>)<sup>T</sup>
<br><br>$$ \Rightarrow $$ A<sup>T</sup> + B<sup>T</sup> = 2B
<br><br>$$ \Rightarrow $$ B = $${{{A^T} + {B^T}} \over 2}$$
<br><br>Now put this in equation (1)
<br><br>So, A + $${{{A^T} + {B^T}} \over 2}$... | mcq | jee-main-2017-online-9th-april-morning-slot |
UJzvMeYq6IQdNHywI6sjl | maths | matrices-and-determinants | transpose-of-a-matrix | Let A = $$\left( {\matrix{
0 & {2q} & r \cr
p & q & { - r} \cr
p & { - q} & r \cr
} } \right).$$ If AA<sup>T</sup> = I<sub>3</sub>, then $$\left| p \right|$$ is : | [{"identifier": "A", "content": "$${1 \\over {\\sqrt 2 }}$$"}, {"identifier": "B", "content": "$${1 \\over {\\sqrt 5 }}$$"}, {"identifier": "C", "content": "$${1 \\over {\\sqrt 6 }}$$"}, {"identifier": "D", "content": "$${1 \\over {\\sqrt 3 }}$$"}] | ["A"] | null | A is orthogonal matrix
<br><br>$$ \Rightarrow $$ 0<sup>2</sup> + p<sup>2</sup> + p<sup>2</sup> = 1
<br><br>$$ \Rightarrow $$ $$\left| p \right| = {1 \over {\sqrt 2 }}$$ | mcq | jee-main-2019-online-11th-january-morning-slot |
K7MM9Pd6ljHdsVeV1i18hoxe66ijvwvdh35 | maths | matrices-and-determinants | transpose-of-a-matrix | The total number of matrices<br/>
$$A = \left( {\matrix{
0 & {2y} & 1 \cr
{2x} & y & { - 1} \cr
{2x} & { - y} & 1 \cr
} } \right)$$<br/>
(x, y $$ \in $$ R,x $$ \ne $$ y) for which A<sup>T</sup>A = 3I<sub>3</sub> is :- | [{"identifier": "A", "content": "3"}, {"identifier": "B", "content": "4"}, {"identifier": "C", "content": "2"}, {"identifier": "D", "content": "6"}] | ["B"] | null | Given A<sup>T</sup>A = 3I<sub>3</sub>
<br><br>$$ \Rightarrow $$ $$\left[ {\matrix{
0 & {2x} & {2x} \cr
{2y} & y & { - y} \cr
1 & { - 1} & 1 \cr
} } \right]\left[ {\matrix{
0 & {2y} & 1 \cr
{2x} & y & { - 1} \cr
{2x} & { - y} & 1 \cr
} } \... | mcq | jee-main-2019-online-9th-april-evening-slot |
oCcGJycx3AUOdA82Pnjgy2xukez6fr99 | maths | matrices-and-determinants | transpose-of-a-matrix | Let a, b, c $$ \in $$ R be all non-zero and satisfy
<br/>a<sup>3</sup> + b<sup>3</sup> + c<sup>3</sup> = 2. If the matrix
<br/><br/>A = $$\left( {\matrix{
a & b & c \cr
b & c & a \cr
c & a & b \cr
} } \right)$$
<br/><br/>satisfies A<sup>T</sup>A = I, then a value of <b>abc</b> c... | [{"identifier": "A", "content": "3"}, {"identifier": "B", "content": "$${1 \\over 3}$$"}, {"identifier": "C", "content": "-$${1 \\over 3}$$"}, {"identifier": "D", "content": "$${2 \\over 3}$$"}] | ["B"] | null | Given, <br>
$${a^3} + {b^3} + {c^3} = 2$$<br><br>
$${A^T}A = I$$<br><br>
$$ \Rightarrow \left[ {\matrix{
a & b & c \cr
b & c & a \cr
c & a & b \cr
} } \right]\left[ {\matrix{
a & b & c \cr
b & c & a \cr
c & a & b \cr
} } \right] = \left[ ... | mcq | jee-main-2020-online-2nd-september-evening-slot |
LQSeGCs3mNm7ThykWi1klrju1ju | maths | matrices-and-determinants | transpose-of-a-matrix | Let M be any 3 $$ \times $$ 3 matrix with entries from the set {0, 1, 2}. The maximum number of such matrices, for which the sum of diagonal elements of M<sup>T</sup>M is seven, is ________. | [] | null | 540 | $$\left[ {\matrix{
a & b & c \cr
d & e & f \cr
g & h & i \cr
} } \right]\left[ {\matrix{
a & d & g \cr
b & e & h \cr
c & f & i \cr
} } \right]$$<br><br>$${a^2} + {b^2} + {c^2} + {d^2} + {e^2} + {f^2} + {g^2} + {h^2} + {i^2} = 7$$<br><br><... | integer | jee-main-2021-online-24th-february-morning-slot |
Be5SSizFcRv9rQls9P1klt99nb8 | maths | matrices-and-determinants | transpose-of-a-matrix | If for the matrix, $$A = \left[ {\matrix{
1 & { - \alpha } \cr
\alpha & \beta \cr
} } \right]$$, $$A{A^T} = {I_2}$$, then the value of $${\alpha ^4} + {\beta ^4}$$ is : | [{"identifier": "A", "content": "3"}, {"identifier": "B", "content": "2"}, {"identifier": "C", "content": "1"}, {"identifier": "D", "content": "4"}] | ["C"] | null | $$\left[ {\matrix{
1 & { - \alpha } \cr
\alpha & \beta \cr
} } \right]\left[ {\matrix{
1 & \alpha \cr
{ - \alpha } & \beta \cr
} } \right] = \left[ {\matrix{
{1 + {\alpha ^2}} & {\alpha - \alpha \beta } \cr
{\alpha - \alpha \beta } & {{\alpha ^2} + {\beta ^2... | mcq | jee-main-2021-online-25th-february-evening-slot |
1l6jekbdu | maths | matrices-and-determinants | transpose-of-a-matrix | <p>Let $$S$$ be the set containing all $$3 \times 3$$ matrices with entries from $$\{-1,0,1\}$$. The total number of matrices $$A \in S$$ such that the sum of all the diagonal elements of $$A^{\mathrm{T}} A$$ is 6 is ____________.</p> | [] | null | 5376 | <p>Sum of all diagonal elements is equal to sum of square of each element of the matrix.</p>
<p>i.e., $$A = \left[ {\matrix{
{{a_1}} & {{a_2}} & {{a_3}} \cr
{{b_1}} & {{b_2}} & {{b_3}} \cr
{{c_1}} & {{c_2}} & {{c_3}} \cr
} } \right]$$</p>
<p>then $${t_r}\,(A\,.\,{A^T})$$</p>
<p>$$ = a_1^2 + a_2^2 + a_3... | integer | jee-main-2022-online-27th-july-morning-shift |
1lgzzvyiu | maths | matrices-and-determinants | transpose-of-a-matrix | <p>Let $$P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right], A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$$ and $$Q=P A P^{T}$$. If $$P^{T} Q^{2007} P=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$$, then $$2 a+... | [{"identifier": "A", "content": "2004"}, {"identifier": "B", "content": "2006"}, {"identifier": "C", "content": "2007"}, {"identifier": "D", "content": "2005"}] | ["D"] | null | $$
\text { Here, } P=\left[\begin{array}{cc}
\frac{\sqrt{3}}{2} & \frac{1}{2} \\
\frac{-1}{2} & \frac{\sqrt{3}}{2}
\end{array}\right], A=\left[\begin{array}{ll}
1 & 1 \\
0 & 1
\end{array}\right]
$$
<br/><br/>$$
\text { Here, } \mathrm{PP}^{\mathrm{T}}=\left[\begin{array}{cc}
\frac{\sqrt{3}}{2} & \frac{1}{2} \\
\frac{-1... | mcq | jee-main-2023-online-8th-april-morning-shift |
lsaob5vy | maths | matrices-and-determinants | transpose-of-a-matrix | If $\mathrm{A}=\left[\begin{array}{cc}\sqrt{2} & 1 \\ -1 & \sqrt{2}\end{array}\right], \mathrm{B}=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right], \mathrm{C}=\mathrm{ABA}^{\mathrm{T}}$ and $\mathrm{X}=\mathrm{A}^{\mathrm{T}} \mathrm{C}^2 \mathrm{~A}$, then $\operatorname{det} \mathrm{X}$ is equa... | [{"identifier": "A", "content": "243"}, {"identifier": "B", "content": "729"}, {"identifier": "C", "content": "27"}, {"identifier": "D", "content": "891"}] | ["B"] | null | <p>The solution involves understanding matrix operations and properties such as multiplication, transpose, and determinant. Given $\mathrm{A}$, $\mathrm{B}$, and that $\mathrm{C} = \mathrm{ABA}^{\mathrm{T}}$, and $\mathrm{X} = \mathrm{A}^{\mathrm{T}} \mathrm{C}^2 \mathrm{A}$, we find $\operatorname{det} \mathrm{X}$ as ... | mcq | jee-main-2024-online-1st-february-morning-shift |
jaoe38c1lsezyqu2 | maths | matrices-and-determinants | transpose-of-a-matrix | <p>Let $$\mathrm{A}$$ be a square matrix such that $$\mathrm{AA}^{\mathrm{T}}=\mathrm{I}$$. Then $$\frac{1}{2} A\left[\left(A+A^T\right)^2+\left(A-A^T\right)^2\right]$$ is equal to</p> | [{"identifier": "A", "content": "$$\\mathrm{A}^2+\\mathrm{A}^{\\mathrm{T}}$$\n"}, {"identifier": "B", "content": "$$\\mathrm{A}^3+\\mathrm{I}$$\n"}, {"identifier": "C", "content": "$$\\mathrm{A}^3+\\mathrm{A}^{\\mathrm{T}}$$\n"}, {"identifier": "D", "content": "$$\\mathrm{A}^2+\\mathrm{I}$$"}] | ["C"] | null | <p>$$\mathrm{AA}^{\mathrm{T}}=\mathrm{I}=\mathrm{A}^{\mathrm{T}} \mathrm{A}$$</p>
<p>On solving given expression, we get</p>
<p>$$\begin{aligned}
& \frac{1}{2} \mathrm{~A}\left[\mathrm{~A}^2+\left(\mathrm{A}^{\mathrm{T}}\right)^2+2 \mathrm{~A} \mathrm{~A}^{\mathrm{T}}+\mathrm{A}^2+\left(\mathrm{A}^{\mathrm{T}}\right)^2... | mcq | jee-main-2024-online-29th-january-morning-shift |
hrCdUD5PJzdt0puJ8q5I7 | maths | parabola | chord-of-contact | Tangents drawn from the point ($$-$$8, 0) to the parabola y<sup>2</sup> = 8x touch the parabola at $$P$$ and $$Q.$$ If F is the focus of the parabola, then the area of the triangle PFQ (in sq. units) is equal to : | [{"identifier": "A", "content": "24"}, {"identifier": "B", "content": "32"}, {"identifier": "C", "content": "48"}, {"identifier": "D", "content": "64"}] | ["C"] | null | Equation of the chord of contact PQ is given by : T=0
<br><br>or T $$ \equiv $$ yy<sub>1</sub> $$-$$ 4(x + x<sub>1</sub>), where (x<sub>1</sub>, y<sub>1</sub><sub></sub>) $$ \equiv $$ ($$-$$8, 0)
<br><br>$$\therefore\,\,\,$$Equation becomes : x = 8
<br><br>& chord of contact is x = 8
<br><br>$$\therefore\,\,\,... | mcq | jee-main-2018-online-15th-april-evening-slot |
7YsDSgdMqevO5bt7 | maths | parabola | chord-of-parabola | If $$a \ne 0$$ and the line $$2bx+3cy+4d=0$$ passes through the points of intersection of the parabolas $${y^2} = 4ax$$ and $${x^2} = 4ay$$, then : | [{"identifier": "A", "content": "$${d^2} + {\\left( {3b - 2c} \\right)^2} = 0$$ "}, {"identifier": "B", "content": "$${d^2} + {\\left( {3b + 2c} \\right)^2} = 0$$ "}, {"identifier": "C", "content": "$${d^2} + {\\left( {2b - 3c} \\right)^2} = 0$$ "}, {"identifier": "D", "content": "$${d^2} + {\\left( {2b + 3c} \\right)^... | ["D"] | null | Solving equations of parabolas
<br><br>$${y^2} = 4ax$$ and $${x^2} = 4ay$$
<br><br>we get $$(0,0)$$ and $$(4a, 4a)$$
<br><br>Substituting in the given equation of line
<br><br>$$2bx+3cy+4d=0,$$
<br><br>we get $$d=0$$
<br><br>and $$2b+3c=0$$ $$ \Rightarrow {d^2} + {\left( {2b + 3c} \right)^2} = 0$$ | mcq | aieee-2004 |
MRfcmYZHlTsDOJCop3OsB | maths | parabola | chord-of-parabola | The length of the chord of the parabola x<sup>2</sup> $$=$$ 4y having equation x – $$\sqrt 2 y + 4\sqrt 2 = 0$$ is - | [{"identifier": "A", "content": "$$8\\sqrt 2 $$"}, {"identifier": "B", "content": "$$6\\sqrt 3 $$"}, {"identifier": "C", "content": "$$3\\sqrt 2 $$"}, {"identifier": "D", "content": "$$2\\sqrt {11} $$"}] | ["B"] | null | x<sup>2</sup> = 4y
<br><br>x $$-$$ $$\sqrt 2 $$y + 4$$\sqrt 2 $$ = 0
<br><br>Solving together we get
<br><br>x<sup>2</sup> = 4$$\left( {{{x + 4\sqrt 2 } \over {\sqrt 2 }}} \right)$$
<br><br>$$\sqrt 2 $$x<sup>2</sup> + 4x + 16$$\sqrt 2 $$
<br><br>$$\sqrt 2 $$x<sup>2</sup> $$-$$ 4x $$-$$ 16$$\sqrt 2 $$ = 0
<br><br>x<sub>... | mcq | jee-main-2019-online-10th-january-evening-slot |
rtn4tbqmzmPGU6Czgf18hoxe66ijvwpvxez | maths | parabola | chord-of-parabola | If one end of a focal chord of the parabola,
y<sup>2</sup> = 16x is at (1, 4), then the length of this focal
chord is : | [{"identifier": "A", "content": "24"}, {"identifier": "B", "content": "20"}, {"identifier": "C", "content": "25"}, {"identifier": "D", "content": "22"}] | ["C"] | null | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734267371/exam_images/fs6tfqb2vks175baakkh.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734267826/exam_images/w5w1xiqap8u9c8okt7io.webp"><img src="https://res.c... | mcq | jee-main-2019-online-9th-april-morning-slot |
YdDyAxUKkRDsCFMrbHjgy2xukf3yz3ju | maths | parabola | chord-of-parabola | Let the latus ractum of the parabola y<sup>2</sup>
= 4x be
the common chord to the circles C<sub>1</sub>
and C<sub>2</sub>
each of them having radius 2$$\sqrt 5 $$. Then, the
distance between the centres of the circles C<sub>1</sub>
and C<sub>2</sub>
is : | [{"identifier": "A", "content": "8"}, {"identifier": "B", "content": "12"}, {"identifier": "C", "content": "$$8\\sqrt 5 $$"}, {"identifier": "D", "content": "$$4\\sqrt 5 $$"}] | ["A"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266972/exam_images/bzjcfypz4zfwsyd69i2a.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2020 (Online) 3rd September Evening Slot Mathematics - Parabola Question 77 English Explanation">
<br>For parabo... | mcq | jee-main-2020-online-3rd-september-evening-slot |
1l545nh5v | maths | parabola | chord-of-parabola | <p>Let PQ be a focal chord of the parabola y<sup>2</sup> = 4x such that it subtends an angle of $${\pi \over 2}$$ at the point (3, 0). Let the line segment PQ be also a focal chord of the ellipse $$E:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$, $${a^2} > {b^2}$$. If e is the eccentricity of the ellipse ... | [{"identifier": "A", "content": "$$1 + \\sqrt 2 $$"}, {"identifier": "B", "content": "$$3 + 2\\sqrt 2 $$"}, {"identifier": "C", "content": "$$1 + 2\\sqrt 3 $$"}, {"identifier": "D", "content": "$$4 + 5\\sqrt 3 $$"}] | ["B"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l5niod7a/d422eadb-ccc2-408b-8388-e3a333afd463/c3bb6e70-04d1-11ed-93b8-936002ac8631/file-1l5niod7b.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l5niod7a/d422eadb-ccc2-408b-8388-e3a333afd463/c3bb6e70-04d1-11ed-93b8-936002ac8631... | mcq | jee-main-2022-online-29th-june-morning-shift |
1l5w1gh8w | maths | parabola | chord-of-parabola | <p>Let PQ be a focal chord of length 6.25 units of the parabola y<sup>2</sup> = 4x. If O is the vertex of the parabola, then 10 times the area (in sq. units) of $$\Delta$$POQ is equal to ___________.</p> | [] | null | 25 | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l6dmy0zj/4987e864-b411-4a25-aaa9-467edd1daa02/d5c9d0f0-132e-11ed-941a-4dd6502f33e3/file-1l6dmy0zk.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l6dmy0zj/4987e864-b411-4a25-aaa9-467edd1daa02/d5c9d0f0-132e-11ed-941a-4dd6502f33e3... | integer | jee-main-2022-online-30th-june-morning-shift |
1l6p2o6sm | maths | parabola | chord-of-parabola | <p>Let the focal chord of the parabola $$\mathrm{P}: y^{2}=4 x$$ along the line $$\mathrm{L}: y=\mathrm{m} x+\mathrm{c}, \mathrm{m}>0$$ meet the parabola at the points M and N. Let the line L be a tangent to the hyperbola $$\mathrm{H}: x^{2}-y^{2}=4$$. If O is the vertex of P and F is the focus of H on the positive ... | [{"identifier": "A", "content": "$$2 \\sqrt{6}$$"}, {"identifier": "B", "content": "$$2 \\sqrt{14}$$"}, {"identifier": "C", "content": "$$4 \\sqrt{6}$$"}, {"identifier": "D", "content": "$$4 \\sqrt{14}$$"}] | ["B"] | null | <p>$$H:{{{x^2}} \over 4} - {{{y^2}} \over 4} = 1$$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l7ssn4pj/b0858943-62f1-4335-ac1e-b62256a92364/16a87a70-2f51-11ed-85dd-19dc023e9ad1/file-1l7ssn4pk.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l7ssn4pj/b0858943-62f1-4335... | mcq | jee-main-2022-online-29th-july-morning-shift |
1ldo71096 | maths | parabola | chord-of-parabola | <p>If the $$x$$-intercept of a focal chord of the parabola $$y^{2}=8x+4y+4$$ is 3, then the length of this chord is equal to ____________.</p> | [] | null | 16 | $$
\begin{aligned}
& y^2=8 x+4 y+4 \\\\
& (y-2)^2=8(x+1) \\\\
& Y^2=4 a X \\\\
& a=2, X=x+1, Y=y-2 \\\\
& \text { focus }(1,2) \\\\
& y-2=m(x-1)
\end{aligned}
$$
<br/><br/>Put $(3,0)$ in the above line $\mathrm{m}=-1$
<br/><br/>Length of focal chord $=16$ | integer | jee-main-2023-online-1st-february-evening-shift |
1lgpxl5r8 | maths | parabola | chord-of-parabola | <p>Let $$\mathrm{PQ}$$ be a focal chord of the parabola $$y^{2}=36 x$$ of length 100 , making an acute angle with the positive $$x$$-axis. Let the ordinate of $$\mathrm{P}$$ be positive and $$\mathrm{M}$$ be the point on the line segment PQ such that PM : MQ = 3 : 1. Then which of the following points does NOT lie on t... | [{"identifier": "A", "content": "$$(6,29)$$"}, {"identifier": "B", "content": "$$(-3,43)$$"}, {"identifier": "C", "content": "$$(3,33)$$"}, {"identifier": "D", "content": "$$(-6,45)$$"}] | ["B"] | null | The given parabola is of the form $$y^2 = 4ax$$. Here, 4a = 36, which means a = 9.
<br><br>Length of focal chord at $(t)=a\left(t+\frac{1}{t}\right)^2=100$
<br><br>Where $a=9$
<br><br>$$
\begin{gathered}
t+\frac{1}{t}= \pm \frac{10}{3} \\\\
\therefore \quad t=3, \frac{1}{3},-3, \frac{-1}{3}
\end{gathered}
$$
<br><br>Si... | mcq | jee-main-2023-online-13th-april-morning-shift |
1lgzyfu4u | maths | parabola | chord-of-parabola | <p>Let $$R$$ be the focus of the parabola $$y^{2}=20 x$$ and the line $$y=m x+c$$ intersect the parabola at two points $$P$$ and $$Q$$.
<br/><br/>Let the point $$G(10,10)$$ be the centroid of the triangle $$P Q R$$. If $$c-m=6$$, then $$(P Q)^{2}$$ is :</p> | [{"identifier": "A", "content": "317"}, {"identifier": "B", "content": "325"}, {"identifier": "C", "content": "346"}, {"identifier": "D", "content": "296"}] | ["B"] | null | $$
y^2=20 x, y=m x+\mathrm{c}
$$
<br/><br/>Put value of $x$
<br/><br/>$$
\begin{aligned}
& y^2=20\left(\frac{y-c}{m}\right) \\\\
& \Rightarrow y^2-\frac{20}{m} y+\frac{20}{m} c=0 .......(i)
\end{aligned}
$$
<br/><br/>Since, centroid $=(10,10)$
<br/><br/>$$
\begin{aligned}
& \text { So, } \frac{y_1+y_2+0}{3}=10 \\\\
& \... | mcq | jee-main-2023-online-8th-april-morning-shift |
jaoe38c1lsfl7tt9 | maths | parabola | chord-of-parabola | <p>Let $$P(\alpha, \beta)$$ be a point on the parabola $$y^2=4 x$$. If $$P$$ also lies on the chord of the parabola $$x^2=8 y$$ whose mid point is $$\left(1, \frac{5}{4}\right)$$, then $$(\alpha-28)(\beta-8)$$ is equal to _________.</p> | [] | null | 192 | <p>Parabola is $$x^2=8 y$$</p>
<p>Chord with mid point $$\left(\mathrm{x}_1, \mathrm{y}_1\right)$$ is $$\mathrm{T}=\mathrm{S}_1$$</p>
<p>$$\begin{aligned}
& \therefore \mathrm{xx}_1-4\left(\mathrm{y}+\mathrm{y}_1\right)=\mathrm{x}_1^2-8 \mathrm{y}_1 \\
& \therefore\left(\mathrm{x}_1, \mathrm{y}_1\right)=\left(1, \frac{... | integer | jee-main-2024-online-29th-january-evening-shift |
lv0vxdso | maths | parabola | chord-of-parabola | <p>Let the length of the focal chord PQ of the parabola $$y^2=12 x$$ be 15 units. If the distance of $$\mathrm{PQ}$$ from the origin is $$\mathrm{p}$$, then $$10 \mathrm{p}^2$$ is equal to __________.</p> | [] | null | 72 | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lwk9u5jo/5c19a59f-8f6e-4ba6-8e8c-779764eeedda/afcaae30-1992-11ef-bba0-ebdcce35fbed/file-1lwk9u5jp.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lwk9u5jo/5c19a59f-8f6e-4ba6-8e8c-779764eeedda/afcaae30-1992-11ef-bba0-ebdcce35fbed... | integer | jee-main-2024-online-4th-april-morning-shift |
lv2er40w | maths | parabola | chord-of-parabola | <p>Let $$P Q$$ be a chord of the parabola $$y^2=12 x$$ and the midpoint of $$P Q$$ be at $$(4,1)$$. Then, which of the following point lies on the line passing through the points $$\mathrm{P}$$ and $$\mathrm{Q}$$ ?</p> | [{"identifier": "A", "content": "$$(3,-3)$$\n"}, {"identifier": "B", "content": "$$\\left(\\frac{1}{2},-20\\right)$$\n"}, {"identifier": "C", "content": "$$(2,-9)$$\n"}, {"identifier": "D", "content": "$$\\left(\\frac{3}{2},-16\\right)$$"}] | ["B"] | null | <p>$$y^2=12 x$$</p>
<p>Chord $$P Q$$ having mid-point $$(x_1, y_1)=(4,1)$$ equation of chord $$P Q$$</p>
<p>$$\begin{aligned}
& T=S_1 \\
& y y_1-12 \frac{\left(x+x_1\right)}{2}=y_1^2-12 x_1 \\
& y-6(x+4)=1-12 \times 4 \\
& y-6 x-24=-47 \\
& y-6 x+23=0
\end{aligned}$$</p>
<p>From option (4) $$x=\frac{1}{2}$$ & $$y=-20$$... | mcq | jee-main-2024-online-4th-april-evening-shift |
lv7v3k4p | maths | parabola | chord-of-parabola | <p>Suppose $$\mathrm{AB}$$ is a focal chord of the parabola $$y^2=12 x$$ of length $$l$$ and slope $$\mathrm{m}<\sqrt{3}$$. If the distance of the chord $$\mathrm{AB}$$ from the origin is $$\mathrm{d}$$, then $$l \mathrm{~d}^2$$ is equal to _________.</p> | [] | null | 108 | <p>Equation of focal chord</p>
<p>$$y-0=\tan \theta .(x-3)$$</p>
<p>Distance from origin</p>
<p>$$\begin{aligned}
& d=\left|\frac{-3 \tan \theta}{\sqrt{1+\tan ^2 \theta}}\right| \\
& I=4 \times 3 \operatorname{cosec}^2 \theta \\
& I. d^2=\frac{9 \tan ^2 \theta}{1+\tan ^2 \theta} \times 12 \operatorname{cosec}^2 \theta ... | integer | jee-main-2024-online-5th-april-morning-shift |
9taDurH0zt2wYzwP | maths | parabola | common-tangent | Two common tangents to the circle $${x^2} + {y^2} = 2{a^2}$$ and parabola $${y^2} = 8ax$$ are : | [{"identifier": "A", "content": "$$x = \\pm \\left( {y + 2a} \\right)$$ "}, {"identifier": "B", "content": "$$y = \\pm \\left( {x + 2a} \\right)$$ "}, {"identifier": "C", "content": "$$x = \\pm \\left( {y + a} \\right)$$ "}, {"identifier": "D", "content": "$$y = \\pm \\left( {x + a} \\right)$$ "}] | ["B"] | null | Any tangent to the parabola $${y^2} = 8ax$$ is
<br><br>$$y = mx + {{2a} \over m}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( i \right)$$
<br><br>If $$(i)$$ is a tangent to the circle, $${x^2} + {y^2} = 2{a^2}$$ then,
<br><br>$$\sqrt {2a} = \pm {{2a} \over {m\sqrt {{m^2} + 1} }}$$
<br><br>$$ \Rightarrow {m^2}\left( {1 + ... | mcq | aieee-2002 |
NtIfshmqF4cgY2Iy | maths | parabola | common-tangent | <b>Given :</b> A circle, $$2{x^2} + 2{y^2} = 5$$ and a parabola, $${y^2} = 4\sqrt 5 x$$.
<br/><b>Statement-1 :</b> An equation of a common tangent to these curves is $$y = x + \sqrt 5 $$.
<p><b>Statement-2 :</b> If the line, $$y = mx + {{\sqrt 5 } \over m}\left( {m \ne 0} \right)$$ is their common tangent, then $$m$$ ... | [{"identifier": "A", "content": "Statement-1 is true; Statement-2 is true; Statement-2 is a correct explanation for Statement-1."}, {"identifier": "B", "content": "Statement-1 is true; Statement-2 is true; Statement-2 is not a correct explanation for Statement-1."}, {"identifier": "C", "content": "Statement-1 is true;... | ["B"] | null | Let common tangent be
<br><br>$$y = mx + {{\sqrt 5 } \over m}$$
<br><br>Since, perpendicular distance from center of the circle to
<br><br>the common tangent is equal to radius of the circle,
<br><br>therefore $${{{{\sqrt 5 } \over m}} \over {\sqrt {1 + {m^2}} }} = \sqrt {{5 \over 2}} $$
<br><br>On squaring both the... | mcq | jee-main-2013-offline |
FxrHLpBJBnUdwvlp1iuzC | maths | parabola | common-tangent | If the common tangents to the parabola, x<sup>2</sup> = 4y and the circle, x<sup>2</sup> + y<sup>2</sup> = 4 intersect at the point P, then the distance of P from the origin, is : | [{"identifier": "A", "content": "$$\\sqrt 2 + 1$$"}, {"identifier": "B", "content": "2(3 + 2 $$\\sqrt 2 $$)"}, {"identifier": "C", "content": "2($$\\sqrt 2 $$ + 1)"}, {"identifier": "D", "content": "3 + 2$$\\sqrt 2 $$"}] | ["B"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734267100/exam_images/dqtfzq9ngy0k8tl3a08o.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2017 (Online) 8th April Morning Slot Mathematics - Parabola Question 104 English Explanation">
<br>Tangent to ... | mcq | jee-main-2017-online-8th-april-morning-slot |
2P3MCSR3dlMypVZh7Vinq | maths | parabola | common-tangent | Two parabolas with a common vertex and with axes along x-axis and $$y$$-axis, respectively intersect each other in the first quadrant. If the length of the latus rectum of each parabola is $$3$$, then the equation of the common tangent to the two parabolas is : | [{"identifier": "A", "content": "4(x + y) + 3 = 0"}, {"identifier": "B", "content": "3(x + y) + 4 = 0"}, {"identifier": "C", "content": "8(2x + y) + 3 = 0"}, {"identifier": "D", "content": "x + 2y + 3 = 0"}] | ["A"] | null | As origin is the only common point to x-axis and y-axis, so origin is the common vertex
<br><br>Let the equation of two of parabolas be y<sup>2</sup> = 4ax and x<sup>2</sup> = 4by
<br><br>Now latus rectum of both parabolas = 3
<br><br>$$\therefore\,\,\,$$ 4a = 4b = 3
<br><br>$$ \Rightarrow $$$$\,\,\,$$ a = b = $${3 ... | mcq | jee-main-2018-online-15th-april-morning-slot |
2ZGfQ5EYzWZNuxj1U6Eng | maths | parabola | common-tangent | Equation of a common tangent to the circle, x<sup>2</sup> + y<sup>2</sup> – 6x = 0 and the parabola, y<sup>2</sup> = 4x is : | [{"identifier": "A", "content": "$$2\\sqrt 3 $$y = 12x + 1"}, {"identifier": "B", "content": "$$\\sqrt 3 $$y = x + 3"}, {"identifier": "C", "content": "$$2\\sqrt 3 $$y = -x - 12"}, {"identifier": "D", "content": "$$\\sqrt 3 $$y = 3x + 1"}] | ["B"] | null | We know,
<br><br>Equation of tangent to the parabola y<sup>2</sup> = 4ax is,
<br><br>y = mx + $${a \over m}$$
<br><br>$$ \therefore $$ Equation of tangent to the parabola y<sup>2</sup> = 4x is,
<br><br>y = mx + $${1 \over m}$$
<br><br>$$ \Rightarrow $$ m<sup>2</sup>x $$-$$ ym + 1 = 0
<br><br>Th... | mcq | jee-main-2019-online-9th-january-morning-slot |
rYubXCr5SvNnWKqiFu3rsa0w2w9jx25iuwn | maths | parabola | common-tangent | If the line ax + y = c, touches both the curves x<sup>2</sup>
+ y<sup>2</sup>
= 1 and y<sup>2</sup>
= 4$$\sqrt 2 $$x , then |c| is equal to : | [{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "$$\\sqrt 2 $$"}, {"identifier": "C", "content": "$${1 \\over {\\sqrt 2 }}$$"}, {"identifier": "D", "content": "$${1 \\over 2}$$"}] | ["B"] | null | Tangent to the curve y<sup>2</sup> = 4$$\sqrt 2$$x is y = mx + $${{\sqrt 2 } \over m}$$<br><br>
It is tangent to the circle x<sup>2</sup> + <sup>y2</sup> = 1<br><br>
$$ \therefore $$ $$\left| {{{\sqrt 2 /m} \over {\sqrt {1 + {m^2}} }}} \right| = 1 \Rightarrow m = \pm 1$$<br><br>
$$ \therefore $$ tangent are y = x + $... | mcq | jee-main-2019-online-10th-april-evening-slot |
hEggXEf1frYDmowbRp3rsa0w2w9jx5c8n12 | maths | parabola | common-tangent | Let P be the point of intersection of the common tangents to the parabola y<sup>2</sup>
= 12x and the hyperbola
8x<sup>2</sup>
– y<sup>2</sup>
= 8. If S and S' denote the foci of the hyperbola where S lies on the positive x-axis then P divides SS'
in a ratio :
| [{"identifier": "A", "content": "14 : 13"}, {"identifier": "B", "content": "13 : 11"}, {"identifier": "C", "content": "5 : 4"}, {"identifier": "D", "content": "2 : 1"}] | ["C"] | null | Let equation of common tangent is y = mx + $${3 \over m}$$<br><br>
$$ \therefore $$ $${\left( {{3 \over m}} \right)^2}$$ = 1, m<sup>2</sup> - 8<br><br>
$$ \Rightarrow {m^4} - 8{m^2} - 9 = 0$$<br><br>
$$ \Rightarrow {m^2} = 9 \Rightarrow m = \pm 3$$<br><br>
$$ \therefore $$ equation of common tangents are y = 3x + 1 &a... | mcq | jee-main-2019-online-12th-april-morning-slot |
7CKGmlezauHN5uhw6S3rsa0w2w9jxad6eun | maths | parabola | common-tangent | The equation of common tangent to the curves y<sup>2</sup>
= 16x and xy = –4, is :
| [{"identifier": "A", "content": "x \u2013 y + 4 = 0"}, {"identifier": "B", "content": "x + y + 4 = 0"}, {"identifier": "C", "content": "x \u2013 2y + 16 = 0"}, {"identifier": "D", "content": "2x \u2013 y + 2 = 0"}] | ["A"] | null | Let the equation of tangent to parabola<br><br>
y<sup>2</sup> = 16x is y = mx + $${4 \over m}$$ ...... (1)<br><br>
It is given that tangent to xy = -4 .........(2)<br><br>
Solving (1) and (2) we get<br><br>
$$x\left( {mx + {4 \over m}} \right) + 4 = 0$$<br><br>
$$ \Rightarrow m{x^2} + {4 \over m}x + 4 = 0$$<br><br>... | mcq | jee-main-2019-online-12th-april-evening-slot |
3X6EHkJyHixQzo6fqxjgy2xukfg6l5e2 | maths | parabola | common-tangent | If the common tangent to the parabolas, <br/>y<sup>2</sup> = 4x and x<sup>2</sup> = 4y also touches the circle, x<sup>2</sup> + y<sup>2</sup> = c<sup>2</sup>,<br/> then c is equal to : | [{"identifier": "A", "content": "$${1 \\over {\\sqrt 2 }}$$"}, {"identifier": "B", "content": "$${1 \\over {2\\sqrt 2 }}$$"}, {"identifier": "C", "content": "$${1 \\over 2}$$"}, {"identifier": "D", "content": "$${1 \\over 4}$$"}] | ["A"] | null | $$y = mx + {1 \over m}$$ (tangent at y<sup>2</sup>
= 4x)
<br><br>y = mx – m<sup>2</sup>
(tangent at x<sup>2</sup>
= 4y)
<br><br>$${1 \over m} = - {m^2}$$ (for common tangent)
<br><br>m<sup>3</sup>
= – 1
<br><br>$$ \Rightarrow $$ m = - 1
<br><br>$$ \therefore $$ Equation of tangent
<br><br>y = –x –1
<br><br>x + y +... | mcq | jee-main-2020-online-5th-september-morning-slot |
rvy2Lk1McfAFgqhIjD1klt9nkb3 | maths | parabola | common-tangent | A line is a common tangent to the circle (x $$-$$ 3)<sup>2</sup> + y<sup>2</sup> = 9 and the parabola y<sup>2</sup> = 4x. If the two points of contact (a, b) and (c, d) are distinct and lie in the first quadrant, then 2(a + c) is equal to _________. | [] | null | 9 | Circle : (x $$-$$ 3)<sup>2</sup> + y<sup>2</sup> = 9<br><br>Parabola : y<sup>2</sup> = 4x<br><br>Let tangent y = mx + $${a \over m}$$<br><br>y = mx + $${1 \over m}$$<br><br>m<sup>2</sup>x $$-$$ my + 1 = 0<br><br>the above line is also tangent to circle<br><br>(x $$-$$ 3)<sup>2</sup> + y<sup>2</sup> = 9<br><br>$$\theref... | integer | jee-main-2021-online-25th-february-evening-slot |
4NuKcpiQ85rmuFolIW1kmkn9dip | maths | parabola | common-tangent | Let L be a tangent line to the parabola y<sup>2</sup> = 4x $$-$$ 20 at (6, 2). If L is also a tangent to the ellipse $${{{x^2}} \over 2} + {{{y^2}} \over b} = 1$$, then the value of b is equal to : | [{"identifier": "A", "content": "20"}, {"identifier": "B", "content": "14"}, {"identifier": "C", "content": "16"}, {"identifier": "D", "content": "11"}] | ["B"] | null | Parabola y<sup>2</sup> = 4x $$-$$ 20<br><br>Tangent at P(6, 2) will be <br><br>$$2y = 4\left( {{{x + 6} \over 2}} \right) - 20$$<br><br>2y = 2x + 12 $$-$$ 20<br><br>2y = 2x $$-$$ 8<br><br>y = x $$-$$ 4<br><br>x $$-$$ y $$-$$ 4 = 0 ....... (1)<br><br>This is also tangent to ellipse $${{{x^2}} \over 2} + {{{y^2}} \over b... | mcq | jee-main-2021-online-17th-march-evening-shift |
1ktkea35m | maths | parabola | common-tangent | A tangent line L is drawn at the point (2, $$-$$4) on the parabola y<sup>2</sup> = 8x. If the line L is also tangent to the circle x<sup>2</sup> + y<sup>2</sup> = a, then 'a' is equal to ___________. | [] | null | 2 | tangent of y<sup>2</sup> = 8x is y = mx + $${2 \over m}$$<br><br>P(2, $$-$$4) $$\Rightarrow$$ $$-$$4 = 2m + $${2 \over m}$$<br><br>$$\Rightarrow$$ m + $${1 \over m}$$ = $$-$$2 $$\Rightarrow$$ m = $$-$$1<br><br>$$\therefore$$ tangent is y = $$-$$x $$-$$2<br><br>$$\Rightarrow$$ x + y + 2 = 0 ...... (1)<br><br>(1) is also... | integer | jee-main-2021-online-31st-august-evening-shift |
1l58ah6pn | maths | parabola | common-tangent | <p>Let the common tangents to the curves $$4({x^2} + {y^2}) = 9$$ and $${y^2} = 4x$$ intersect at the point Q. Let an ellipse, centered at the origin O, has lengths of semi-minor and semi-major axes equal to OQ and 6, respectively. If e and l respectively denote the eccentricity and the length of the latus rectum of th... | [] | null | 4 | <p>Let y = mx + c is the common tangent</p>
<p>So $$c = {1 \over m} = \pm \,{3 \over 2}\sqrt {1 + {m^2}} \Rightarrow {m^2} = {1 \over 3}$$</p>
<p>So equation of common tangents will be $$y = \pm \,{1 \over {\sqrt 3 }}x \pm \,\sqrt 3 $$, which intersects at Q($$-$$3, 0)</p>
<p>Major axis and minor axis of ellipse are... | integer | jee-main-2022-online-26th-june-morning-shift |
1l5aisjco | maths | parabola | common-tangent | <p>If $$y = {m_1}x + {c_1}$$ and $$y = {m_2}x + {c_2}$$, $${m_1} \ne {m_2}$$ are two common tangents of circle $${x^2} + {y^2} = 2$$ and parabola y<sup>2</sup> = x, then the value of $$8|{m_1}{m_2}|$$ is equal to :</p> | [{"identifier": "A", "content": "$$3 + 4\\sqrt 2 $$"}, {"identifier": "B", "content": "$$ - 5 + 6\\sqrt 2 $$"}, {"identifier": "C", "content": "$$ - 4 + 3\\sqrt 2 $$"}, {"identifier": "D", "content": "$$7 + 6\\sqrt 2 $$"}] | ["C"] | null | <p>Let tangent to $${y^2} = x$$ be</p>
<p>$$y = mx + {1 \over {4m}}$$</p>
<p>For it being tangent to circle.</p>
<p>$$\left| {{{{1 \over 4}m} \over {\sqrt {1 + {m^2}} }}} \right| = \sqrt 2 $$</p>
<p>$$ \Rightarrow 32{m^4} + 32{m^2} - 1 = 0$$</p>
<p>$$ \Rightarrow {m^2} = {{ - 32 \pm \sqrt {{{(32)}^2} + 4(32)} } \over {... | mcq | jee-main-2022-online-25th-june-morning-shift |
1l5c0w908 | maths | parabola | common-tangent | <p>Let x<sup>2</sup> + y<sup>2</sup> + Ax + By + C = 0 be a circle passing through (0, 6) and touching the parabola y = x<sup>2</sup> at (2, 4). Then A + C is equal to ___________.</p> | [{"identifier": "A", "content": "16"}, {"identifier": "B", "content": "88/5"}, {"identifier": "C", "content": "72"}, {"identifier": "D", "content": "$$-$$8"}] | ["A"] | null | For tangent to parabola $y=x^{2}$ at $(2,4)$
<br/><br/>
$$
\left.\frac{d y}{d x}\right|_{(2,4)}=4
$$
<br/><br/>
Equation of tangent is
$$
y-4=4(x-2)
$$
<br/><br/>
$\Rightarrow 4 x-y-4=0$
<br/><br/>
Family of circle can be given by
<br/><br/>
$(x-2)^{2}+(y-4)^{2}+\lambda(4 x-y-4)=0$
<br/><br/>
As it passes through $(0,... | mcq | jee-main-2022-online-24th-june-morning-shift |
1l6hyskfa | maths | parabola | common-tangent | <p>The equation of a common tangent to the parabolas $$y=x^{2}$$ and $$y=-(x-2)^{2}$$ is</p> | [{"identifier": "A", "content": "$$y=4(x-2)$$"}, {"identifier": "B", "content": "$$y=4(x-1)$$"}, {"identifier": "C", "content": "$$y=4(x+1)$$"}, {"identifier": "D", "content": "$$y=4(x+2)$$"}] | ["B"] | null | <p>Equation of tangent of slope $$m$$ to $$y$$ $$= x^2$$</p>
<p>$$y = mx - {1 \over 4}{m^2}$$</p>
<p>Equation of tangent of slope $$m$$ to $$y = - {(x - 2)^2}$$</p>
<p>$$y = m(x - 2) + {1 \over 4}{m^2}$$</p>
<p>If both equation represent the same line</p>
<p>$${1 \over 4}{m^2} - 2m = - {1 \over 4}{m^2}$$</p>
<p>$$m =... | mcq | jee-main-2022-online-26th-july-evening-shift |
1l6np9p51 | maths | parabola | common-tangent | <p>Two tangent lines $$l_{1}$$ and $$l_{2}$$ are drawn from the point $$(2,0)$$ to the parabola $$2 \mathrm{y}^{2}=-x$$. If the lines $$l_{1}$$ and $$l_{2}$$ are also tangent to the circle $$(x-5)^{2}+y^{2}=r$$, then 17r is equal to ___________.</p> | [] | null | 9 | <p>Given : $${y^2} = {{ - x} \over 2}$$</p>
<p>$$\eqalign{
& T \equiv y = mx - {1 \over {8m}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \downarrow (2,0) \cr} $$</p>
<p>$$ \Rightarrow {m^2} = {1 \over {16}} \Rightarrow m = \, \pm \,{1 \over 4}$$</p>
<p>Tangents are $$y = {1 \over 4}x - {1 \over 2}... | integer | jee-main-2022-online-28th-july-evening-shift |
ldqwkapa | maths | parabola | common-tangent | Let $A$ be a point on the $x$-axis. Common tangents are drawn from $A$ to the curves $x^2+y^2=8$ and $y^2=16 x$. If one of these tangents touches the two curves at $Q$ and $R$, then $(Q R)^2$ is equal to : | [{"identifier": "A", "content": "76"}, {"identifier": "B", "content": "81"}, {"identifier": "C", "content": "64"}, {"identifier": "D", "content": "72"}] | ["D"] | null | <p>Let a tangent on $${y^2} = 16x$$ be $$y = mx + {4 \over m}$$</p>
<p>For common to $${x^2} + {y^2} = 8$$</p>
<p>$${4 \over m} = 2\sqrt 2 (1 + {m^2})$$</p>
<p>$$ \Rightarrow {2 \over {{m^2}}} = 1 + {m^2} \Rightarrow m = \, \pm 1$$</p>
<p>Taking one of the tangent $$y = x + 4$$</p>
<p>Point of tangency with $${y^2} = 4... | mcq | jee-main-2023-online-30th-january-evening-shift |
1ldv2j1e5 | maths | parabola | common-tangent | <p>The distance of the point $$(6,-2\sqrt2)$$ from the common tangent $$\mathrm{y=mx+c,m > 0}$$, of the curves $$x=2y^2$$ and $$x=1+y^2$$ is :</p> | [{"identifier": "A", "content": "$$\\frac{1}{3}$$"}, {"identifier": "B", "content": "5"}, {"identifier": "C", "content": "$$\\frac{14}{3}$$"}, {"identifier": "D", "content": "5$$\\sqrt3$$"}] | ["B"] | null | $$
\begin{aligned}
& y^2=\frac{x}{2} \Rightarrow \text { tangent } y=m x+\frac{1}{8 m} \\\\
& y^2=x-1 \Rightarrow \text { tangent } y=m(x-1)+\frac{1}{4 m} \\\\
& \text { For common tangent } \frac{1}{8 m}=-m+\frac{1}{4 m} \\\\
& \Rightarrow 1=-8 m^2+2 \\\\
& \because m>0 \Rightarrow m=\frac{1}{2 \sqrt{2}} \\\\
& \Right... | mcq | jee-main-2023-online-25th-january-morning-shift |
1lgxwffpa | maths | parabola | common-tangent | <p>Let a common tangent to the curves $${y^2} = 4x$$ and $${(x - 4)^2} + {y^2} = 16$$ touch the curves at the points P and Q. Then $${(PQ)^2}$$ is equal to __________.</p> | [] | null | 32 | Tangent of slope $m$ to the parabola
<br/><br/>$y^2=4 x$ is given by $y=m x+\frac{1}{m}$ and <br/><br/>Tangent of slope $m$ to the circle $(x-4)^2+y^2=16$ is given by
<br/><br/>$$
y=m(x-4) \pm 4 \sqrt{1+m^2}
$$
<br/><br/>For common tangent
<br/><br/>$$
\begin{aligned}
& \frac{1}{m}=-4 m \pm 4 \sqrt{1+m^2} \\\\
& \Righ... | integer | jee-main-2023-online-10th-april-morning-shift |
lsaq1mxh | maths | parabola | common-tangent | Let the line $\mathrm{L}: \sqrt{2} x+y=\alpha$ pass through the point of the intersection $\mathrm{P}$ (in the first quadrant) of the circle $x^2+y^2=3$ and the parabola $x^2=2 y$. Let the line $\mathrm{L}$ touch two circles $\mathrm{C}_1$ and $\mathrm{C}_2$ of equal radius $2 \sqrt{3}$. If the centres $Q_1$ and $Q_2$ ... | [] | null | 72 | <p>$x^2+y^2=3$ and $x^2=2 y$</p>
$y^2+2 y-3=0 $
<br/><br/>$\Rightarrow(y+3)(y-1)=0$
<br/><br/>$y=-3$ (Rejected) or $y=1$
<br/><br/>For $\mathrm{y}=1, \mathrm{x}=\sqrt{2} \Rightarrow P(\sqrt{2}, 1)$
<br/><br/>$p$ lies on the line
<br/><br/>$$
\begin{aligned}
& \sqrt{2} x+y=\alpha \\\\
& \sqrt{2}(\sqrt{2})+1=\al... | integer | jee-main-2024-online-1st-february-morning-shift |
lvc57b5a | maths | parabola | common-tangent | <p>Let $$C$$ be the circle of minimum area touching the parabola $$y=6-x^2$$ and the lines $$y=\sqrt{3}|x|$$. Then, which one of the following points lies on the circle $$C$$ ?</p> | [{"identifier": "A", "content": "$$(1,2)$$\n"}, {"identifier": "B", "content": "$$(2,2)$$\n"}, {"identifier": "C", "content": "$$(1,1)$$\n"}, {"identifier": "D", "content": "$$(2,4)$$"}] | ["D"] | null | <p>Let centre be (0, k)</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lwd1ocxr/d26061ca-43e1-4f4a-a0fc-77af60f91e91/491160f0-1599-11ef-acc0-2dadf5616f59/file-1lwd1ocxs.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lwd1ocxr/d26061ca-43e1-4f4a-a0fc-77af60f91e91/491160f0... | mcq | jee-main-2024-online-6th-april-morning-shift |
evpXasRCTfr92zs0 | maths | parabola | locus | Let $$P$$ be the point $$(1, 0)$$ and $$Q$$ a point on the parabola $${y^2} = 8x$$. The locus of mid point of $$PQ$$ is : | [{"identifier": "A", "content": "$${y^2} - 4x + 2 = 0$$ "}, {"identifier": "B", "content": "$${y^2} + 4x + 2 = 0$$"}, {"identifier": "C", "content": "$${x^2} + 4y + 2 = 0$$"}, {"identifier": "D", "content": "$${x^2} - 4y + 2 = 0$$"}] | ["A"] | null | $$P = \left( {1,0} \right)\,\,Q = \left( {h,k} \right)$$ Such that $${k^2} = 8h$$
<br><br>Let $$\left( {\alpha ,\beta } \right)$$ be the midpoint of $$PQ$$
<br><br>$$\alpha = {{h + 1} \over 2},\,\,\,\beta = {{k + 0} \over 2}$$
<br><br>$$ \therefore $$ $$2\alpha - 1 = h\,\,\,\,\,\,2\beta = k.$$
<br><br>$${\left( {... | mcq | aieee-2005 |
QFprF6TCLSKFhBp8 | maths | parabola | locus | If two tangents drawn from a point $$P$$ to the parabola $${y^2} = 4x$$ are at right angles, then the locus of $$P$$ is | [{"identifier": "A", "content": "$$2x+1=0$$"}, {"identifier": "B", "content": "$$x=-1$$ "}, {"identifier": "C", "content": "$$2x-1=0$$ "}, {"identifier": "D", "content": "$$x=1$$ "}] | ["B"] | null | The locus of perpendicular tangents is directrix
<br><br>i.e., $$x=-1$$ | mcq | aieee-2010 |
2KV25TyV87muR2Xt | maths | parabola | locus | Let $$O$$ be the vertex and $$Q$$ be any point on the parabola, $${{x^2} = 8y}$$. If the point $$P$$ divides the line segment $$OQ$$ internally in the ratio $$1:3$$, then locus of $$P$$ is : | [{"identifier": "A", "content": "$${y^2} = 2x$$ "}, {"identifier": "B", "content": "$${{x^2} = 2y}$$ "}, {"identifier": "C", "content": "$${{x^2} = y}$$"}, {"identifier": "D", "content": "$${y^2} = x$$ "}] | ["B"] | null | <p>Let the coordinates of Q and P be (x<sub>1</sub>, y<sub>1</sub>) and (h, k) respectively.</p>
<p>$$\because$$ Q lies on x<sup>2</sup> = 8y,</p>
<p>$$\therefore$$ x$$_1^2$$ = 8y ....... (1)</p>
<p>Again, P divides OQ internally in the ratio 1 : 3.</p>
<p>$$\therefore$$ $$h = {{{x_1} + 0} \over 4} = {{{x_1}} \over 4}$... | mcq | jee-main-2015-offline |
8r6EX9Sd1qXSMJx8mk7k9k2k5gypkn5 | maths | parabola | locus | The locus of a point which divides the line
segment joining the point (0, –1) and a point on
the parabola, x<sup>2</sup> = 4y, internally in the ratio
1 : 2, is : | [{"identifier": "A", "content": "9x<sup>2</sup> \u2013 3y = 2"}, {"identifier": "B", "content": "4x<sup>2</sup> \u2013 3y = 2"}, {"identifier": "C", "content": "x<sup>2</sup> \u2013 3y = 2"}, {"identifier": "D", "content": "9x<sup>2</sup> \u2013 12y = 8"}] | ["D"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266223/exam_images/w7aqioqqabs5dzmfr7a7.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2020 (Online) 8th January Morning Slot Mathematics - Parabola Question 82 English Explanation">
<br>Take point P... | mcq | jee-main-2020-online-8th-january-morning-slot |
Zs5uWLenVyBwNLYaxY1klrhcogv | maths | parabola | locus | The locus of the mid-point of the line segment joining the focus of the parabola y<sup>2</sup> = 4ax to a
moving point of the parabola, is another parabola whose directrix is : | [{"identifier": "A", "content": "x = 0"}, {"identifier": "B", "content": "x = - $${a \\over 2}$$"}, {"identifier": "C", "content": "x = a"}, {"identifier": "D", "content": "x = $${a \\over 2}$$"}] | ["A"] | null | Given, equation of parabola $$\Rightarrow$$ y<sup>2</sup> = 4ax<br><br>Focus = S(a, 0)<br><br>Let any point on the parabola be P(at<sup>2</sup>, 2at).<br><br><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1kxof1w7l/840d5c1f-ce31-4701-93d3-208583e5d683/a5091c10-66ee-11ec-9866-0df874a38238/file-1kxof1w7m... | mcq | jee-main-2021-online-24th-february-morning-slot |
1krrulg3k | maths | parabola | locus | Let P be a variable point on the parabola $$y = 4{x^2} + 1$$. Then, the locus of the mid-point of the point P and the foot of the perpendicular drawn from the point P to the line y = x is : | [{"identifier": "A", "content": "$${(3x - y)^2} + (x - 3y) + 2 = 0$$"}, {"identifier": "B", "content": "$$2{(3x - y)^2} + (x - 3y) + 2 = 0$$"}, {"identifier": "C", "content": "$${(3x - y)^2} + 2(x - 3y) + 2 = 0$$"}, {"identifier": "D", "content": "$$2{(x - 3y)^2} + (3x - y) + 2 = 0$$"}] | ["B"] | null | Given, parabola $$y = 4{x^2} + 1$$<br><br><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l09usogd/0e8d490d-75bf-4d50-a3ef-e9477504eeb8/1b3947d0-9a51-11ec-b1c5-39b9b722e9af/file-1l09usoge.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l09usogd/0e8d490d-75bf-4d50-a3ef-e9477504ee... | mcq | jee-main-2021-online-20th-july-evening-shift |
1ktfzhxju | maths | parabola | locus | If two tangents drawn from a point P to the <br/>parabola y<sup>2</sup> = 16(x $$-$$ 3) are at right angles, then the locus of point P is : | [{"identifier": "A", "content": "x + 3 = 0"}, {"identifier": "B", "content": "x + 1 = 0"}, {"identifier": "C", "content": "x + 2 = 0"}, {"identifier": "D", "content": "x + 4 = 0"}] | ["B"] | null | Locus is directrix of parabola<br><br>x $$-$$ 3 + 4 = 0 $$\Rightarrow$$ x + 1 = 0. | mcq | jee-main-2021-online-27th-august-evening-shift |
ldqwyq2x | maths | parabola | locus | The parabolas : $a x^2+2 b x+c y=0$ and $d x^2+2 e x+f y=0$ intersect on the line $y=1$. If $a, b, c, d, e, f$ are positive real numbers and $a, b, c$ are in G.P., then : | [{"identifier": "A", "content": "$\\frac{d}{a}, \\frac{e}{b}, \\frac{f}{c}$ are in A.P."}, {"identifier": "B", "content": "$\\frac{d}{a}, \\frac{e}{b}, \\frac{f}{c}$ are in G.P."}, {"identifier": "C", "content": "$d, e, f$ are in A.P."}, {"identifier": "D", "content": "$d, e, f$ are in G.P."}] | ["A"] | null | <p>Let point of intersection be ($$\alpha,1$$)</p>
<p>$$\alpha x^2+2b\alpha+c=0$$ ..... (i)</p>
<p>and $$d\alpha^2+2e\alpha+f=0$$ .... (ii)</p>
<p>$$\Rightarrow a\alpha^2+2\sqrt{ac}\alpha+c=0$$ ($$\because$$ $$b^2=ac$$)</p>
<p>$${\left( {\sqrt a \alpha + \sqrt c } \right)^2} = 0$$</p>
<p>$$\alpha = - \sqrt {{c \over... | mcq | jee-main-2023-online-30th-january-evening-shift |
1ldu41vqc | maths | parabola | locus | <p>The equations of two sides of a variable triangle are $$x=0$$ and $$y=3$$, and its third side is a tangent to the parabola $$y^2=6x$$. The locus of its circumcentre is :</p> | [{"identifier": "A", "content": "$$4{y^2} - 18y - 3x - 18 = 0$$"}, {"identifier": "B", "content": "$$4{y^2} + 18y + 3x + 18 = 0$$"}, {"identifier": "C", "content": "$$4{y^2} - 18y + 3x + 18 = 0$$"}, {"identifier": "D", "content": "$$4{y^2} - 18y - 3x + 18 = 0$$"}] | ["C"] | null | Third side of triangle
<br><br>
$t y=x+\frac{3}{2} t^{2}$<br><br>
<img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lef4vkb3/455513c9-54ce-4923-8412-8892bb374d49/794a91f0-b263-11ed-a6d1-894f1caef997/file-1lef4vkb4.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lef4vkb3/455513c9-5... | mcq | jee-main-2023-online-25th-january-evening-shift |
dn3PoBQdOLPq820v | maths | parabola | normal-to-parabola | The normal at the point$$\left( {bt_1^2,2b{t_1}} \right)$$ on a parabola meets the parabola again in the point $$\left( {bt_2^2,2b{t_2}} \right)$$, then : | [{"identifier": "A", "content": "$${t_2} = {t_1} + {2 \\over {{t_1}}}$$ "}, {"identifier": "B", "content": "$${t_2} = -{t_1} - {2 \\over {{t_1}}}$$"}, {"identifier": "C", "content": "$${t_2} = -{t_1} + {2 \\over {{t_1}}}$$"}, {"identifier": "D", "content": "$${t_2} = {t_1} - {2 \\over {{t_1}}}$$"}] | ["B"] | null | Equation of the normal to a parabola $${y^2} = 4bx$$ at point
<br><br>$$\left( {bt_1^2,2b{t_1}} \right)$$ is $$y = - {t_1}x + 2b{t_1} + bt_1^3$$
<br><br>As given, it also passes through $$\left( {bt_2^2,2b{t_2}} \right)$$ then
<br><br>$$2b{t_2} = {t_1}bt_2^2 + 2b{t_1} + bt_1^3$$
<br><br>$$2{t_2} - 2{t_1} = - {t_1}... | mcq | aieee-2003 |
rF6OFC4tBXIjjAbO | maths | parabola | normal-to-parabola | Let $$P$$ be the point on the parabola, $${{y^2} = 8x}$$ which is at a minimum distance from the centre $$C$$ of the circle, $${x^2} + {\left( {y + 6} \right)^2} = 1$$. Then the equation of the circle, passing through $$C$$ and having its centre at $$P$$ is: | [{"identifier": "A", "content": "$${{x^2} + {y^2} - {x \\over 4} + 2y - 24 = 0}$$ "}, {"identifier": "B", "content": "$${{x^2} + {y^2} - 4x + 9y + 18 = 0}$$ "}, {"identifier": "C", "content": "$${{x^2} + {y^2} - 4x + 8y + 12 = 0}$$"}, {"identifier": "D", "content": "$${{x^2} + {y^2} - x + 4y - 12 = 0}$$"}] | ["C"] | null | Minimum distance $$ \Rightarrow $$ perpendicular distance
<br><br>$$E{q^n}$$ of normal at $$p\left( {2{t^2},\,4t} \right)$$
<br><br>$$y = - tx + 4t + 2{t^3}$$
<br><br>It passes through $$C\left( {0, - 6} \right) \Rightarrow {t^3} + 2t + 3 = 0$$
<br><br>$$ \Rightarrow t = - 1$$
<br><br><img class="question-image" sr... | mcq | jee-main-2016-offline |
IDKgSlFmiu6IKjtRmTg5G | maths | parabola | normal-to-parabola | P and Q are two distinct points on the parabola, y<sup>2</sup> = 4x, with parameters t and t<sub>1</sub> respectively. If the normal at P passes through Q, then the minimum value of $$t_1^2$$ is : | [{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "4"}, {"identifier": "C", "content": "6"}, {"identifier": "D", "content": "8"}] | ["D"] | null | t<sub>1</sub> = $$-$$ t $$-$$ $${2 \over t}$$
<br><br>$$t_1^2$$ = t<sup>2</sup> + $${4 \over {{t^2}}}$$ + 4
<br><br>t<sup>2</sup> + $${4 \over {{t^2}}}$$ $$ \ge $$ 2$$\sqrt {{t^2}.{4 \over {{t^2}}}} = 4$$
<br><br>Minimum value of $$t_1^2$$ = 8 | mcq | jee-main-2016-online-10th-april-morning-slot |
aYPG8LoQabTT8eGQ2ua7k | maths | parabola | normal-to-parabola | If y = mx + c is the normal at a point on the parabola y<sup>2</sup> = 8x whose focal distance is 8 units, then $$\left| c \right|$$ is equal to : | [{"identifier": "A", "content": "$$2\\sqrt 3 $$ "}, {"identifier": "B", "content": "$$8\\sqrt 3 $$"}, {"identifier": "C", "content": "$$10\\sqrt 3 $$"}, {"identifier": "D", "content": "$$16\\sqrt 3 $$"}] | ["C"] | null | c = $$-$$ 29m $$-$$ 9m<sup>3</sup>
<br><br>a = 2
<br><br>Given (at<sup>2</sup> $$-$$ a)<sup>2</sup> + 4a<sup>2</sup>t<sup>2</sup> = 64
<br><br>$$ \Rightarrow $$ (a(t<sup>2</sup> + 1)) = 8
<br><br>$$ \Rightarrow $$ t<sup>2</sup> + 1 = 4 $$ \Rightarrow $$ t<sup>2</sup> = 3
<br><br>$... | mcq | jee-main-2017-online-9th-april-morning-slot |
YjXpdEyhqPAl1IA4 | maths | parabola | normal-to-parabola | Tangent and normal are drawn at P(16, 16) on the parabola y<sup>2</sup> = 16x, which intersect the axis of the
parabola at A and B, respectively. If C is the centre of the circle through the points P, A and B and $$\angle $$CPB =
$$\theta $$, then a value of tan$$\theta $$ is : | [{"identifier": "A", "content": "$${4 \\over 3}$$"}, {"identifier": "B", "content": "$${1 \\over 2}$$"}, {"identifier": "C", "content": "2"}, {"identifier": "D", "content": "3"}] | ["C"] | null | <img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263693/exam_images/be9bgy7e2oqgaj3q4lr8.webp" loading="lazy" alt="JEE Main 2018 (Offline) Mathematics - Parabola Question 108 English Explanation 1">
<br><br>As equation of tangent PA at (x<sub>1</sub>, y<sub>1</sub>) on the parabo... | mcq | jee-main-2018-offline |
xHGKs3J3uPk1BI64yCR9N | maths | parabola | normal-to-parabola | If the parabolas y<sup>2</sup> = 4b(x – c) and y<sup>2</sup> = 8ax have a common normal, then which on of the following is a valid choice for the ordered triad (a, b, c)? | [{"identifier": "A", "content": "(1, 1, 3)"}, {"identifier": "B", "content": "(1, 1, 0)"}, {"identifier": "C", "content": "$$\\left( {{1 \\over 2},2,0} \\right)$$"}, {"identifier": "D", "content": "$$\\left( {{1 \\over 2},2,3} \\right)$$"}] | ["A"] | null | Normal to the two given curves are
<br><br>y = m(x – c) – 2bm – bm<sup>3</sup>,
<br><br>y = mx – 4am – 2am<sup>3</sup>
<br><br>If they have a common normal, then
<br><br>(c + 2b)m + bm<sup>3</sup>
= 4am + 2am<sup>3</sup>
<br><br>$$ \Rightarrow $$ (4a – c – 2b) m = (b – 2a)m<sup>3</sup>
<br><br>$$ \Rightarrow $$ (4a – ... | mcq | jee-main-2019-online-10th-january-morning-slot |
R82TmCwmYj5NHZ0Vju18hoxe66ijvwvfhji | maths | parabola | normal-to-parabola | The area (in sq. units) of the smaller of the two
circles that touch the parabola, y<sup>2 </sup> = 4x at the point
(1, 2) and the x-axis is :- | [{"identifier": "A", "content": "$$4\\pi \\left( {3 +\\sqrt 2 } \\right)$$"}, {"identifier": "B", "content": "$$8\\pi \\left( {2 - \\sqrt 2 } \\right)$$"}, {"identifier": "C", "content": "$$8\\pi \\left( {3 - 2\\sqrt 2 } \\right)$$"}, {"identifier": "D", "content": "$$4\\pi \\left( {2 - \\sqrt 2 } \\right)$$"}] | ["C"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264518/exam_images/csbm6jz6ovhaimejwhpu.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 9th April Evening Slot Mathematics - Parabola Question 88 English Explanation">
<br><br>Equation o... | mcq | jee-main-2019-online-9th-april-evening-slot |
qrVANARoXQ1CZObJIUjgy2xukf49t0l9 | maths | parabola | normal-to-parabola | If the tangent to the curve, y = e<sup>x</sup>
at a point
(c, e<sup>c</sup>) and the normal to the parabola, y<sup>2</sup> = 4x
at the point (1, 2) intersect at the same point on
the x-axis, then the value of c is ________ .
| [] | null | 4 | For $$y = {e^x}$$<br><br>$${{dy} \over {dx}} = {e^x}$$<br><br>$${\left. {{{dy} \over {dx}}} \right|_{x = c}} = {e^c}$$<br><br>Tangent is $$y - {e^c} = {e^c}(x - c)$$<br><br>Put y = 0, x = c$$ - $$1.........(i)<br><br>For y<sup>2</sup> = 4x<br><br>$$2y{{dy} \over {dx}} = 4 \Rightarrow {\left. {{{ - dx} \over {dy}}} \rig... | integer | jee-main-2020-online-3rd-september-evening-slot |
52WEiaRzrURiv0g4XY1kmhw1mcp | maths | parabola | normal-to-parabola | If the three normals drawn to the parabola, y<sup>2</sup> = 2x pass through the point (a, 0) a $$\ne$$ 0, then 'a' must be greater than : | [{"identifier": "A", "content": "$${1 \\over 2}$$"}, {"identifier": "B", "content": "1"}, {"identifier": "C", "content": "$$-$$1"}, {"identifier": "D", "content": "$$-$$$${1 \\over 2}$$"}] | ["B"] | null | Let the equation of the normal is <br><br>y = mx $$-$$ 2am $$-$$ am<sup>3</sup><br><br>here 4a = 2 $$ \Rightarrow $$ a = $${1 \over 2}$$<br><br>y = mx $$-$$ m $$-$$ $${1 \over 2}$$m<sup>3</sup><br><br>It passing through A(a, 0) then<br><br>0 = am $$-$$ m $$-$$ $${1 \over 2}$$m<sup>3</sup><br><br>m = 0, a $$-$$ 1 $$-$$ ... | mcq | jee-main-2021-online-16th-march-morning-shift |
1krpzucdd | maths | parabola | normal-to-parabola | Let the tangent to the parabola S : y<sup>2</sup> = 2x at the point P(2, 2) meet the x-axis at Q and normal at it meet the parabola S at the point R. Then the area (in sq. units) of the triangle PQR is equal to : | [{"identifier": "A", "content": "$${{25} \\over 2}$$"}, {"identifier": "B", "content": "$${{35} \\over 2}$$"}, {"identifier": "C", "content": "$${{15} \\over 2}$$"}, {"identifier": "D", "content": "25"}] | ["A"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266392/exam_images/kfqvzltdhtj2yuoqxjth.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 20th July Morning Shift Mathematics - Parabola Question 63 English Explanation"><br><br>Tangent at... | mcq | jee-main-2021-online-20th-july-morning-shift |
1ktely03s | maths | parabola | normal-to-parabola | A tangent and a normal are drawn at the point P(2, $$-$$4) on the parabola y<sup>2</sup> = 8x, which meet the directrix of the parabola at the points A and B respectively. If Q(a, b) is a point such that AQBP is a square, then 2a + b is equal to : | [{"identifier": "A", "content": "$$-$$16"}, {"identifier": "B", "content": "$$-$$18"}, {"identifier": "C", "content": "$$-$$12"}, {"identifier": "D", "content": "$$-$$20"}] | ["A"] | null | <p>Given, parabola</p>
<p>$${y^2} = 8x$$ ...... (i)</p>
<p>Equation of tangent at $$P(2, - 4)$$ is</p>
<p>$$ - 4y = 4(x + 2)$$</p>
<p>or, $$x + y + 2 = 0$$ ..... (ii)</p>
<p>and Equation of normal to the parabola is</p>
<p>$$x - y + C = 0$$</p>
<p>$$\therefore$$ Normal passes through $$(2, - 4)$$</p>
<p>$$\therefore$$ ... | mcq | jee-main-2021-online-27th-august-morning-shift |
1kto6jqrb | maths | parabola | normal-to-parabola | Consider the parabola with vertex $$\left( {{1 \over 2},{3 \over 4}} \right)$$ and the directrix $$y = {1 \over 2}$$. Let P be the point where the parabola meets the line $$x = - {1 \over 2}$$. If the normal to the parabola at P intersects the parabola again at the point Q, then (PQ)<sup>2</sup> is equal to : | [{"identifier": "A", "content": "$${{75} \\over 8}$$"}, {"identifier": "B", "content": "$${{125} \\over {16}}$$"}, {"identifier": "C", "content": "$${{25} \\over 2}$$"}, {"identifier": "D", "content": "$${{15} \\over 2}$$"}] | ["B"] | null | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1kwopy91l/b12d9de1-4cc2-4e41-a53c-978e58e1dcd9/0cc11390-534d-11ec-9cbb-695a838b20fb/file-1kwopy91m.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1kwopy91l/b12d9de1-4cc2-4e41-a53c-978e58e1dcd9/0cc11390-534d-11ec-9cbb-695a838b20fb/fi... | mcq | jee-main-2021-online-1st-september-evening-shift |
1l589slgl | maths | parabola | normal-to-parabola | <p>Let the normal at the point on the parabola y<sup>2</sup> = 6x pass through the point (5, $$-$$8). If the tangent at P to the parabola intersects its directrix at the point Q, then the ordinate of the point Q is :</p> | [{"identifier": "A", "content": "$$-$$3"}, {"identifier": "B", "content": "$$-$$$${{9} \\over 4}$$"}, {"identifier": "C", "content": "$$-$$$${{5} \\over 2}$$"}, {"identifier": "D", "content": "$$-$$2"}] | ["B"] | null | <p>Let P(at<sup>2</sup>, 2at) where a = $${3 \over 2}$$</p>
<p>T : yt = x + at<sup>2</sup> So point Q is $$\left( { - a,\,at - {a \over t}} \right)$$</p>
<p>N : y = $$-$$tx + 2at + at<sup>3</sup> passes through (5, $$-$$8)</p>
<p>$$-$$8 = $$-$$5t + 3t + $${3 \over 2}$$t<sup>3</sup></p>
<p>$$\Rightarrow$$ 3t<sup>3</sup>... | mcq | jee-main-2022-online-26th-june-morning-shift |
1ldr6xehk | maths | parabola | normal-to-parabola | <p>If $$\mathrm{P}(\mathrm{h}, \mathrm{k})$$ be a point on the parabola $$x=4 y^{2}$$, which is nearest to the point $$\mathrm{Q}(0,33)$$, then the distance of $$\mathrm{P}$$ from the directrix of the parabola $$\quad y^{2}=4(x+y)$$ is equal to :</p> | [{"identifier": "A", "content": "8"}, {"identifier": "B", "content": "2"}, {"identifier": "C", "content": "6"}, {"identifier": "D", "content": "4"}] | ["C"] | null | <p>Equation of normal</p>
<p>$$y = - tx + 2.{1 \over {16}}t + {1 \over {16}}{t^3}$$</p>
<p>$$33 = {t \over 8} + {{{t^3}} \over {16}}$$</p>
<p>$$ \Rightarrow {t^3} + 2t = 528$$</p>
<p>$$t = 8$$</p>
<p>$$(a{t^2},2at) = (4,1)$$</p>
<p>Distance from $$x = - 2$$</p> | mcq | jee-main-2023-online-30th-january-morning-shift |
1lgsw758t | maths | parabola | normal-to-parabola | <p>Let the tangent to the parabola $$\mathrm{y}^{2}=12 \mathrm{x}$$ at the point $$(3, \alpha)$$ be perpendicular to the line $$2 x+2 y=3$$. Then the square of distance of the point $$(6,-4)$$ from the normal to the hyperbola $$\alpha^{2} x^{2}-9 y^{2}=9 \alpha^{2}$$ at its point $$(\alpha-1, \alpha+2)$$ is equal to __... | [] | null | 116 | $\because \mathrm{P}(3, \alpha)$ lies on $\mathrm{y}^2=12 \mathrm{x}$
<br/><br/>$\Rightarrow \alpha= \pm 6$
<br/><br/>$$
\text { But, }\left.\frac{\mathrm{dy}}{\mathrm{dx}}\right|_{(3, \alpha)}=\frac{6}{\alpha}=1 \Rightarrow \alpha=6(\alpha=-6 \text { reject })
$$
<br/><br/>Now, hyperbola $\frac{x^2}{9}-\frac{y^2}{36}... | integer | jee-main-2023-online-11th-april-evening-shift |
lsbki99q | maths | parabola | normal-to-parabola | If the shortest distance of the parabola $y^2=4 x$ from the centre of the circle $x^2+y^2-4 x-16 y+64=0$ is $\mathrm{d}$, then $\mathrm{d}^2$ is equal to : | [{"identifier": "A", "content": "16"}, {"identifier": "B", "content": "24"}, {"identifier": "C", "content": "20"}, {"identifier": "D", "content": "36"}] | ["C"] | null | <p>Equation of normal to parabola</p>
<p>$$\mathrm{y=m x-2 m-m^3}$$</p>
<p>this normal passing through center of circle $$(2,8)$$</p>
<p>$$\begin{aligned}
& 8=2 \mathrm{~m}-2 \mathrm{~m}-\mathrm{m}^3 \\
& \mathrm{~m}=-2
\end{aligned}$$</p>
<p>So point $$\mathrm{P}$$ on parabola $$\Rightarrow\left(\mathrm{am}^2,-2 \math... | mcq | jee-main-2024-online-27th-january-morning-shift |
1lgypxtlo | maths | parabola | pair-of-tangents | <p>The ordinates of the points P and $$\mathrm{Q}$$ on the parabola with focus $$(3,0)$$ and directrix $$x=-3$$ are in the ratio $$3: 1$$. If $$\mathrm{R}(\alpha, \beta)$$ is the point of intersection of the tangents to the parabola at $$\mathrm{P}$$ and $$\mathrm{Q}$$, then $$\frac{\beta^{2}}{\alpha}$$ is equal to ___... | [] | null | 16 | $$
\begin{aligned}
& \text { Give parabola is : } y^2=12 x \quad(\because a=3) \\\\
& \text { So, } \mathrm{P} \equiv\left(a t_1^2, 2 a t_1\right) \\\\
& \mathrm{Q} \equiv\left(a t_2^2, 2 a t_2\right) \\\\
& \text { So, point } \mathrm{R}(\alpha, \beta) \equiv\left(a t_1 t_2, a\left(t_1+t_2\right)\right) \\\\
& \equiv(... | integer | jee-main-2023-online-8th-april-evening-shift |
AspiOe4ZyUJUoiRTgB09N | maths | parabola | position-of-point-and-chord-joining-of-two-points | Let A(4, $$-$$ 4) and B(9, 6) be points on the parabola, y<sup>2</sup> = 4x. Let C be chosen on the arc AOB of the parabola, where O is the origin, such that the area of $$\Delta $$ACB is maximum. Then, the area (in sq. units) of $$\Delta $$ACB, is : | [{"identifier": "A", "content": "$$31{1 \\over 4}$$"}, {"identifier": "B", "content": "$$30{1 \\over 2}$$"}, {"identifier": "C", "content": "32"}, {"identifier": "D", "content": "$$31{3 \\over 4}$$"}] | ["A"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264184/exam_images/dtyf9rtine6siiyeobjo.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 9th January Evening Slot Mathematics - Parabola Question 98 English Explanation">
<br><br>$$\Del... | mcq | jee-main-2019-online-9th-january-evening-slot |
aqj3lPSrYhSaqBB1SDp7y | maths | parabola | position-of-point-and-chord-joining-of-two-points | If the area of the triangle whose one vertex is at the vertex of the parabola, y<sup>2</sup> + 4(x – a<sup>2</sup>) = 0 and the othertwo vertices are the points of intersection of the parabola and y-axis, is 250 sq. units, then a value of 'a' is : | [{"identifier": "A", "content": "$$5\\sqrt 5 $$"}, {"identifier": "B", "content": "$${\\left( {10} \\right)^{2/3}}$$"}, {"identifier": "C", "content": "$$5\\left( {{2^{1/3}}} \\right)$$"}, {"identifier": "D", "content": "5"}] | ["D"] | null | Vertex is (a<sup>2</sup>, 0)
<br><br>y<sup>2</sup> $$=$$ $$-$$(x $$-$$ a<sup>2</sup>) and x $$=$$ 0 $$ \Rightarrow $$ (0, $$ \pm $$ 2a)
<br><br>Area of triangle is $$ = {1 \over 2}.$$4a.(a<sup>2</sup>) = 250
<br><br>$$ \Rightarrow $$ a<sup>3</sup> = 125 or a = 5 | mcq | jee-main-2019-online-11th-january-evening-slot |
ElZ6UyxucQlC2uTZyf0kk | maths | parabola | position-of-point-and-chord-joining-of-two-points | The maximum area (in sq. units) of a rectangle having its base on the x-axis and its other two vertices on the parabola, y = 12 – x<sup>2</sup> such that the rectangle lies inside the parabola, is :
| [{"identifier": "A", "content": "36"}, {"identifier": "B", "content": "20$$\\sqrt 2 $$"}, {"identifier": "C", "content": "18$$\\sqrt 3 $$"}, {"identifier": "D", "content": "32"}] | ["D"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734267052/exam_images/mnslvvuc5ia6lxqcjf12.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 12th January Morning Slot Mathematics - Parabola Question 94 English Explanation">
<br>f (a) = 2... | mcq | jee-main-2019-online-12th-january-morning-slot |
INEmnq65GsBAbnWypAlve | maths | parabola | position-of-point-and-chord-joining-of-two-points | Let P(4, –4) and Q(9, 6) be two points on the parabola, y<sup>2</sup> = 4x and let x be any point on the arc POQ of this parabola, where O is the vertex of this parabola, such that the area of $$\Delta $$PXQ is maximum. Then this maximum area (in sq. units) is : | [{"identifier": "A", "content": "$${{625} \\over 4}$$"}, {"identifier": "B", "content": "$${{125} \\over 4}$$"}, {"identifier": "C", "content": "$${{75} \\over 2}$$"}, {"identifier": "D", "content": "$${{125} \\over 2}$$"}] | ["B"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266547/exam_images/ql9pnjqathvtux4uc0ar.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 12th January Morning Slot Mathematics - Parabola Question 93 English Explanation">
<br>y<sup>2</... | mcq | jee-main-2019-online-12th-january-morning-slot |
b93Qm4pBuW9a6iMQF5jgy2xukez5j28t | maths | parabola | position-of-point-and-chord-joining-of-two-points | The area (in sq. units) of an equilateral triangle
inscribed in the parabola y<sup>2</sup> = 8x, with one of
its vertices on the vertex of this parabola, is : | [{"identifier": "A", "content": "$$256\\sqrt 3 $$"}, {"identifier": "B", "content": "$$64\\sqrt 3 $$"}, {"identifier": "C", "content": "$$128\\sqrt 3 $$"}, {"identifier": "D", "content": "$$192\\sqrt 3 $$"}] | ["D"] | null | Let A = (2t<sup>2</sup>
, 4t)
<br><br>B = (2t<sup>2</sup>
, -4t) (by symmetry as equilateral
triangle)
<br><img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263641/exam_images/fiy4gcor4i4z43cmqerx.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2020... | mcq | jee-main-2020-online-2nd-september-evening-slot |
L76uaJ1Go15PoOPVJXjgy2xukf0pks9k | maths | parabola | position-of-point-and-chord-joining-of-two-points | Let P be a point on the parabola, y<sup>2</sup>
= 12x and
N be the foot of the perpendicular drawn from
P on the axis of the parabola. A line is now
drawn through the mid-point M of PN, parallel
to its axis which meets the parabola at Q. If the
y-intercept of the line NQ is $${4 \over 3}$$,
then : | [{"identifier": "A", "content": "MQ = $${1 \\over 3}$$"}, {"identifier": "B", "content": "PN = 4"}, {"identifier": "C", "content": "PN = 3"}, {"identifier": "D", "content": "MQ = $${1 \\over 4}$$"}] | ["D"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265153/exam_images/uwoptt2srzgkzl7wy0ci.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2020 (Online) 3rd September Morning Slot Mathematics - Parabola Question 78 English Explanation">
<br><br>Let P ... | mcq | jee-main-2020-online-3rd-september-morning-slot |
ir2l5yqzikZtHZQSfz1klrlp86w | maths | parabola | position-of-point-and-chord-joining-of-two-points | If P is a point on the parabola y = x<sup>2</sup> + 4 which is closest to the straight line y = 4x $$-$$ 1, then the co-ordinates of P are : | [{"identifier": "A", "content": "($$-$$2, 8)"}, {"identifier": "B", "content": "(2, 8)"}, {"identifier": "C", "content": "(1, 5)"}, {"identifier": "D", "content": "(3, 13)"}] | ["B"] | null | Given, curve y = x<sup>2</sup> + 4<br><br>and, line y = 4x $$-$$ 1<br><br>Here, y = x<sup>2</sup> + 4<br><br><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1kxoae8y6/3b0cfa8b-90f4-4e91-ba41-db601a79f96b/6ef08ee0-66dc-11ec-8eea-2b236925a651/file-1kxoae8y7.png?format=png" data-orsrc="https://app-content.... | mcq | jee-main-2021-online-24th-february-evening-slot |
CA3h2nvK6MgPA2sGlp1klt7a0gw | maths | parabola | position-of-point-and-chord-joining-of-two-points | The shortest distance between the line x $$-$$ y = 1 and the curve x<sup>2</sup> = 2y is : | [{"identifier": "A", "content": "0"}, {"identifier": "B", "content": "$${1 \\over 2{\\sqrt 2 }}$$"}, {"identifier": "C", "content": "$${1 \\over {\\sqrt 2 }}$$"}, {"identifier": "D", "content": "$${1 \\over 2}$$"}] | ["B"] | null | Shortest distance must be along common normal<br><br><img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264020/exam_images/ipwtluvs8xeh3hc4yv0v.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 25th February Evening Shift Mathematics - Par... | mcq | jee-main-2021-online-25th-february-evening-slot |
1krrwj01w | maths | parabola | position-of-point-and-chord-joining-of-two-points | If the point on the curve y<sup>2</sup> = 6x, nearest to the point $$\left( {3,{3 \over 2}} \right)$$ is ($$\alpha$$, $$\beta$$), then 2($$\alpha$$ + $$\beta$$) is equal to _____________. | [] | null | 9 | Let, $$P \equiv \left( {{3 \over 2}{t^2},3t} \right)$$ is on the curve.<br><br>Normal at point P<br><br>$$tx + y = 3t + {3 \over 2}{t^3}$$<br><br>Passes through $$\left( {3,{3 \over 2}} \right)$$<br><br>$$ \Rightarrow 3t + {3 \over 2} = 3t + {3 \over 2}{t^3}$$<br><br>$$ \Rightarrow {t^3} = 1 \Rightarrow t = 1$$<br><br>... | integer | jee-main-2021-online-20th-july-evening-shift |
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