data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence with a total of $20$ terms, the sum of all terms is $75$, and the sum of the even terms is $25$, determine the common difference $d$."
}
] | math | {
"style": "rule",
"ground_truth": "-2.5"
} | {
"index": 13900,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence {a_n} with the sum of its first n terms denoted as S_n, and given that a_1008 > 0 and a_1007 + a_1008 < 0, find the positive integer value(s) of n that satisfy S_nS_{n+1} < 0."
}
] | math | {
"style": "rule",
"ground_truth": "2014"
} | {
"index": 13901,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence, the sum of the first four terms is 26, the sum of the last four terms is 110, and the sum of all terms in the sequence is 187. Determine the total number of terms in the sequence."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 13902,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic-geometric sequence $\\{ a_{n} \\}$ that satisfies $a\\_1 + a\\_3 = 10$, $a\\_2 + a\\_4 = 5$, find the maximum value of the product $a\\_1 a\\_2 \\ldots a\\_n$."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 13903,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic-geometric sequence $\\{a\\_n\\}$ with a sum of the first $n$ terms denoted as $S\\_n$ and a common ratio of $\\frac{3}{2}$.\n(1) If $S\\_4 = \\frac{65}{24}$, find $a\\_1$;\n(2) If $a\\_1=2$, $c\\_n = \\frac{1}{2}a\\_n + bn$, and $c\\_2$, $c\\_4$, $c\\_5$ for... | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{16}"
} | {
"index": 13904,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic-geometric sequence $\\{a\\_n\\}$, where $a\\_1 + a\\_3 = 10$ and $a\\_4 + a\\_6 = \\frac{5}{4}$, find its fourth term and the sum of the first five terms."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{31}{2}"
} | {
"index": 13905,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic-geometric sequence $\\{a_n\\}$ with the sum of its first $n$ terms denoted as $S_n$, if $a_3 - 4a_2 + 4a_1 = 0$, find the value of $\\frac{S_8}{S_4}$."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 13906,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic-geometric sequence ${a_n}$ with the sum of its first $n$ terms denoted as $S_n$, and it is known that $\\frac{S_6}{S_3} = -\\frac{19}{8}$ and $a_4 - a_2 = -\\frac{15}{8}$. Find the value of $a_3$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{4}"
} | {
"index": 13907,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic-geometric sequence {$a_n$} with the first term as $\\frac{4}{3}$ and a common ratio of $- \\frac{1}{3}$. The sum of its first n terms is represented by $S_n$. If $A ≤ S_{n} - \\frac{1}{S_{n}} ≤ B$ holds true for any n∈N*, find the minimum value of B - A."
... | math | {
"style": "rule",
"ground_truth": "\\frac{59}{72}"
} | {
"index": 13908,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $$C: \\frac {x^{2}}{a^{2}}+ \\frac {y^{2}}{b^{2}}=1$$ and a hyperbola $$\\frac {x^{2}}{4-v}+ \\frac {y^{2}}{1-v}=1 (1<v<4)$$ share a common focus. A line $l$ passes through the right vertex B of the ellipse C and intersects the parabola $y^2=2x$ at points P and... | math | {
"style": "rule",
"ground_truth": "\\frac {1}{2}"
} | {
"index": 13909,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $$C: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$$ with eccentricity $$\\frac{\\sqrt{3}}{2}$$, and the distance from its left vertex to the line $x + 2y - 2 = 0$ is $$\\frac{4\\sqrt{5}}{5}$$.\n\n(Ⅰ) Find the equation of ellipse C; \n(Ⅱ) Suppose line $l... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"index": 13910,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $$\\frac {x^{2}}{a^{2}}$$ + $$\\frac {y^{2}}{b^{2}}$$ = 1 with its right focus F, a line passing through the origin O intersects the ellipse C at points A and B. If |AF| = 2, |BF| = 4, and the eccentricity of the ellipse C is $$\\frac {\\sqrt {7}}{3}$$, calcula... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 13911,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $(C)$: $\\frac{x^{2}}{3m} + \\frac{y^{2}}{m} = 1 (m > 0)$ with the length of its major axis being $2\\sqrt{6}$, and $O$ is the coordinate origin.\n(I) Find the equation and eccentricity of the ellipse $(C)$;\n(II) Let moving line $(l)$ intersect with the $y$-ax... | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"index": 13912,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $(C)$: $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a > b > 0)$, the minor axis is $2 \\sqrt{3}$, and the eccentricity $e= \\frac{1}{2}$,\n(1) Find the standard equation of ellipse $(C)$;\n(2) If $F_{1}$ and $F_{2}$ are the left and right foci of ellipse $(C)... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 13913,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C$ with its center at the origin and its foci on the $x$-axis, and its eccentricity equal to $\\frac{1}{2}$. One of its vertices is exactly the focus of the parabola $x^{2}=8\\sqrt{3}y$.\n\n(Ⅰ) Find the standard equation of the ellipse $C$;\n\n(Ⅱ) If the line ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 13914,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C$ with its left and right foci at $F_{1}(-\\sqrt{3},0)$ and $F_{2}(\\sqrt{3},0)$, respectively, and the ellipse passes through the point $(-1, \\frac{\\sqrt{3}}{2})$.\n\n(Ⅰ) Find the equation of the ellipse $C$; \n(Ⅱ) Given a fixed point $A(1, \\frac{1}{2})$... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 13915,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C$ with one focus at $F_{1}(2,0)$ and the corresponding directrix $x=8$, and eccentricity $e=\\frac{1}{2}$.\n\n$(1)$ Find the equation of the ellipse $C$;\n\n$(2)$ Find the length of the chord cut from the ellipse $C$ by a line passing through the other focus ... | math | {
"style": "rule",
"ground_truth": "\\frac{48}{7}"
} | {
"index": 13916,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C$: $\\dfrac {x^{2}}{a^{2}}+ \\dfrac {y^{2}}{b^{2}}=1(a > b > 0)$ with its left and right foci being $F_{1}$ and $F_{2}$ respectively, and its eccentricity is $\\dfrac {1}{2}$. Let $M$ be any point on the ellipse and the perimeter of $\\triangle MF_{1}F_{2}$ e... | math | {
"style": "rule",
"ground_truth": "\\dfrac { \\sqrt {15}}{3}"
} | {
"index": 13917,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C$: $\\dfrac{x^{2}}{a^{2}} + \\dfrac{y^{2}}{b^{2}} = 1$ passes through points $M(2,0)$ and $N(0,1)$.\n\n$(1)$ Find the equation of ellipse $C$ and its eccentricity;\n\n$(2)$ A line $y=kx (k \\in \\mathbb{R}, k \\neq 0)$ intersects ellipse $C$ at points $A$ and... | math | {
"style": "rule",
"ground_truth": "\\dfrac{8}{5}"
} | {
"index": 13918,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C$: $\\frac{x^2}{16}+\\frac{y^2}{4}=1$, with the left and right foci being $F_{1}$ and $F_{2}$, respectively. Line $l$ intersects the ellipse at points $A$ and $B$, where the chord $AB$ is bisected by the point $(\\sqrt{3},\\frac{\\sqrt{3}}{2})$. \n$(1)$ Find... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{15}"
} | {
"index": 13919,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C$: $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$ passes through the point $(1, \\frac{2\\sqrt{3}}{3})$, with its left and right foci being $F_1$ and $F_2$ respectively. The chord formed by the intersection of the circle $x^2 + y^2 = 2$ and the line $x... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"index": 13920,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C$: $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$, where the upper vertex of $C$ is $A$, and the two foci are $F_{1}$ and $F_{2}$, with an eccentricity of $\\frac{1}{2}$. A line passing through $F_{1}$ and perpendicular to $AF_{2}$ intersects $C$ at po... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 13921,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C$: $\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1(a>b>0)$ with an eccentricity of $\\frac{\\sqrt{3}}{2}$ and a length of the minor axis of $4$. <br/>$(1)$ Find the equation of the ellipse; <br/>$(2)$ A chord passing through $P(2,1)$ divides $P$ in half. Find the equati... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"index": 13922,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C$: $\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1(a>b>0)$ with eccentricity $\\frac{2\\sqrt{2}}{3}$, the line $y=\\frac{1}{2}$ intersects $C$ at points $A$ and $B$, where $|AB|=3\\sqrt{3}$. \n$(1)$ Find the equation of $C$; \n$(2)$ Let the left and right foci of $C$ ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 13923,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C$: $\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1\\left(a \\gt b \\gt 0\\right)$ with the left and right foci $F_{1}$ and $F_{2}$ respectively. Let $F_{1}$ be the center of a circle with radius $|F_{1}F_{2}|$, intersecting the $y$-axis at points $A$ and $B$ (both point... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3} - 1"
} | {
"index": 13924,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C$: $\\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ passing through the point $(0,4)$, with an eccentricity of $\\frac{3}{5}$\n1. Find the equation of $C$;\n2. Find the length of the line segment intercepted by $C$ on the line passing through poi... | math | {
"style": "rule",
"ground_truth": "\\frac{41}{5}"
} | {
"index": 13925,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C$: $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with a focal length of $2$, and point $Q( \\frac{a^{2}}{ \\sqrt{a^{2}-b^{2}}},0)$ on the line $l$: $x=2$.\n(1) Find the standard equation of the ellipse $C$;\n(2) Let $O$ be the coordinate origin, ... | math | {
"style": "rule",
"ground_truth": "\\frac{ \\sqrt{2}}{2}"
} | {
"index": 13926,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C$: $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with an eccentricity of $\\frac{3}{5}$ and a minor axis length of $8$,\n\n(1) Find the standard equation of the ellipse $C$;\n\n(2) Let $F_{1}$ and $F_{2}$ be the left and right foci of the ellipse... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"index": 13927,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C$: $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with its upper vertex at $(0,2)$ and an eccentricity of $\\frac{\\sqrt{5}}{3}$.\n(1) Find the equation of ellipse $C$;\n(2) From a point $P$ on the ellipse $C$, draw two tangent lines to the circle... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"index": 13928,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C$: $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\\frac{1}{2}$, a circle with the origin as its center and the short axis of the ellipse as its radius is tangent to the line $\\sqrt{7}x-\\sqrt{5}y+12=0$.\n(1) Find the equa... | math | {
"style": "rule",
"ground_truth": "-\\frac{12}{7}"
} | {
"index": 13929,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C$: $\\frac{{x}^{2}}{3}+\\frac{{y}^{2}}{{b}^{2}}=1\\left(b \\gt 0\\right)$ with the right focus $F$, two lines passing through $F$ are tangent to the circle $O$: $x^{2}+y^{2}=r^{2}(r \\gt 0)$ at points $A$ and $B$, forming a right triangle $\\triangle ABF$. It... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 13930,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C$: $\\frac{{x}^{2}}{3}+{y}^{2}=1$ with left focus and right focus as $F_{1}$ and $F_{2}$ respectively, the line $y=x+m$ intersects $C$ at points $A$ and $B$. Determine the value of $m$ such that the area of $\\triangle F_{1}AB$ is twice the area of $\\triangl... | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{2}}{3}"
} | {
"index": 13931,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C$: $\\frac{{x}^{2}}{3}+{y}^{2}=1$ with left focus and right focus as $F_{1}$ and $F_{2}$ respectively. The line $y=x+m$ intersects $C$ at points $A$ and $B$. If the area of $\\triangle F_{1}AB$ is twice the area of $\\triangle F_{2}AB$, find the value of $m$.... | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{2}}{3}"
} | {
"index": 13932,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C: \\dfrac{x^2}{a^2} + \\dfrac{y^2}{b^2} = 1 \\left( a > b > 0 \\right)$ with eccentricity $\\dfrac{\\sqrt{3}}{2}$, and the length of its minor axis is $2$.\n\n(I) Find the standard equation of the ellipse $C$.\n\n(II) Suppose a tangent line $l$ of the circle ... | math | {
"style": "rule",
"ground_truth": "\\dfrac{5}{4}"
} | {
"index": 13933,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$ whose right focus is $F$, and a circle with center $F$: $x^2 + y^2 - 4x - 32 = 0$ that is tangent to the right directrix of the ellipse, find the eccentricity of the ellipse."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 13934,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$, its left and right foci are $F_1$ and $F_2$ respectively. Point $P(1, \\frac{\\sqrt{2}}{2})$ is on the ellipse, and $|PF_1| + |PF_2| = 2\\sqrt{2}$.\n\n$(1)$ Find the standard equation of ellipse $C$; \n... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 13935,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ ($a > b > 0$) with a vertex coordinate at $B(0,1)$, and the eccentricity of the ellipse being $\\frac{\\sqrt{3}}{2}$,\n$(I)$ find the equation of the ellipse;\n$(II)$ let point $Q$ be a point on ellipse $C$ below the... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{7}"
} | {
"index": 13936,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ ($a > b > 0$) with an eccentricity of $\\frac{\\sqrt{5}}{5}$, and the equation of the right directrix is $x=5$.\n\n(1) Find the equation of the ellipse; \n(2) A line $l$ with a slope of 1 passes through the right fo... | math | {
"style": "rule",
"ground_truth": "\\frac{16\\sqrt{10}}{9}"
} | {
"index": 13937,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C: \\frac{x^{2}}{4} + y^{2} = 1$, with $O$ being the origin of coordinates, and a line $l$ intersects the ellipse $C$ at points $A$ and $B$, and $\\angle AOB = 90^{\\circ}$.\n\n(Ⅰ) If the line $l$ is parallel to the x-axis, find the area of $\\triangle AOB$; ... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{5}}{5}"
} | {
"index": 13938,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C: \\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, whose left and right foci are $F_{1}$ and $F_{2}$ respectively, and the top vertex is $B$. If the perimeter of $\\triangle BF_{1}F_{2}$ is $6$, and the distance from point $F_{1}$ to the line $BF... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 13939,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C: \\frac{y^{2}}{a^{2}}+ \\frac{x^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\\frac{\\sqrt{2}}{2}$ and the sum of the distances from a point on the ellipse to the two foci is $2\\sqrt{2}$. A line $l$ with slope $k(k\\neq 0)$ passes through the upper fo... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{6}}{16}"
} | {
"index": 13940,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C:\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1 \\left( a > b > 0 \\right)$ with eccentricity $\\dfrac{1}{2}$, and it passes through the point $\\left( 1,\\dfrac{3}{2} \\right)$. If point $M\\left( x_{0},y_{0} \\right)$ is on the ellipse $C$, then the point $N\\left( ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 13941,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C:\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1(a > b > 0)$ with its right focus $F$ lying on the line $2x-y-2=0$, where $A$ and $B$ are the left and right vertices of $C$, and $|AF|=3|BF|$.<br/>$(1)$ Find the standard equation of $C$;<br/>$(2)$ A line $l$ passing throu... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{4}"
} | {
"index": 13942,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C:\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1(a>b>0)$ with left and right foci $F_{1}$ and $F_{2}$, and a point $P(1,\\frac{{\\sqrt{2}}}{2})$ on the ellipse, satisfying $|PF_{1}|+|PF_{2}|=2\\sqrt{2}$.<br/>$(1)$ Find the standard equation of the ellipse $C$;<br/>$(2)$ ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 13943,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C:\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1(a>b>0)$ with left and right foci $F_{1}$, $F_{2}$. There exists a point $M$ in the first quadrant of ellipse $C$ such that $|MF_{1}|=|F_{1}F_{2}|$. Line $F_{1}M$ intersects the $y$-axis at point $A$, and $F_{2}A$ bisects $... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5} - 1}{2}"
} | {
"index": 13944,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C:\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with eccentricity $\\frac{1}{2}$, a circle $\\odot E$ with center at the origin and radius equal to the minor axis of the ellipse is tangent to the line $x-y+\\sqrt{6}=0$. <br/>$(1)$ Find the equati... | math | {
"style": "rule",
"ground_truth": "\\frac{6\\sqrt{6}}{11}"
} | {
"index": 13945,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C:\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with foci $F_{1}$ and $F_{2}$, point $A$ lies on $C$, point $B$ lies on the $y$-axis, and satisfies $\\overrightarrow{AF_{1}}⊥\\overrightarrow{BF_{1}}$, $\\overrightarrow{AF_{2}}=\\frac{2}{3}\\overr... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{5}"
} | {
"index": 13946,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C:\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with foci $F_{1}$ and $F_{2}$, point $A$ lies on $C$, point $B$ lies on the $y$-axis, and satisfies $\\overrightarrow{A{F}_{1}}⊥\\overrightarrow{B{F}_{1}}$, $\\overrightarrow{A{F}_{2}}=\\frac{2}{3}\... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{5}"
} | {
"index": 13947,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C:\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with left focus $F$ and a chord perpendicular to the major axis of length $6\\sqrt{2}$, a line passing through point $P(2,1)$ with slope $-1$ intersects $C$ at points $A$ and $B$, where $P$ is the m... | math | {
"style": "rule",
"ground_truth": "6\\sqrt{2} + 6"
} | {
"index": 13948,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $C:\\frac{{{x}^{2}}}{{{a}^{2}}}+\\frac{{{y}^{2}}}{{{b}^{2}}}=1\\left( a > b > 0 \\right)$ with left and right focal points ${F}_{1},{F}_{2}$, and a point $P\\left( 1,\\frac{\\sqrt{2}}{2} \\right)$ on the ellipse such that $\\vert P{F}_{1}\\vert+\\vert P{F}_{2}\... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 13949,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $E$ with eccentricity $\\frac{\\sqrt{2}}{2}$: $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1(a > b > 0)$ that passes through point $A(1, \\frac{\\sqrt{2}}{2})$.\n\n$(1)$ Find the equation of ellipse $E$; \n$(2)$ If a line $l$: $y= \\frac{\\sqrt{2}}{2}x+m$ that ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 13950,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $E$: $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ ($a > b > 0$) whose left focus $F_1$ coincides with the focus of the parabola $y^2 = -4x$, and the eccentricity of ellipse $E$ is $\\frac{\\sqrt{2}}{2}$. A line $l$ with a non-zero slope passes through point $M(m,0... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 13951,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $E$: $\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1(a>b>0)$ with left focus $F_{1}$ and right focus $F_{2}$, and the focal distance $F_{1}F_{2}$ is $2$. A line passing through $F_{1}$ intersects the ellipse $E$ at points $A$ and $B$, and the perimeter of $\\triangle ABF_... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{15}}{7}"
} | {
"index": 13952,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $E$: $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\\frac{\\sqrt{3}}{2}$ and a minor axis length of $2$.\n1. Find the equation of the ellipse $E$;\n2. A line $l$ is tangent to a circle $C$: $x^{2}+y^{2}=r^{2}(0 < r < b)$ at ... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{5}}{5}"
} | {
"index": 13953,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $E: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$, there is a point $M(2,1)$ inside it. Two lines $l_1$ and $l_2$ passing through $M$ intersect the ellipse $E$ at points $A$, $C$ and $B$, $D$ respectively, and satisfy $\\overrightarrow{AM}=\\lambda \\ove... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"index": 13954,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $E: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 \\left(a > b > 0\\right)$ that passes through the point $(0,1)$, and its eccentricity is $\\frac{\\sqrt{3}}{2}$.\n\n$(1)$ Find the standard equation of the ellipse $E$;\n\n$(2)$ Suppose a line $l: y = \\frac{1}{2}x + ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{2}"
} | {
"index": 13955,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $E: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ ($a > b > 0$) whose left focus $F_1$ coincides with the focus of the parabola $y^2 = -4x$, and the eccentricity of ellipse $E$ is $\\frac{\\sqrt{2}}{2}$. A line $l$ with a non-zero slope passes through point $M(m, 0)... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 13956,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $E: \\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1(a>b>0)$ with an eccentricity of $\\frac{\\sqrt{2}}{2}$ and upper vertex at B. Point P is on E, point D is at (0, -2b), and the maximum area of △PBD is $\\frac{3\\sqrt{2}}{2}$.\n(I) Find the equation of E;\n(II) I... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 13957,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $E:\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with a major axis length of $4$, and the point $P(1,\\frac{3}{2})$ lies on the ellipse $E$. <br/>$(1)$ Find the equation of the ellipse $E$; <br/>$(2)$ A line $l$ passing through the right focus $F$... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 13958,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $G$: $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with an eccentricity of $\\frac{\\sqrt{6}}{3}$, a right focus at $(2 \\sqrt{2},0)$, and a line $l$ with a slope of $1$ intersecting the ellipse $G$ at points $A$ and $B$. An isosceles triangle is c... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"index": 13959,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $M$ with its axes of symmetry being the coordinate axes, and its eccentricity is $\\frac{\\sqrt{2}}{2}$, and one of its foci is at $(\\sqrt{2}, 0)$.\n\n$(1)$ Find the equation of the ellipse $M$; \n$(2)$ Suppose a line $l$ intersects the ellipse $M$ at points ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 13960,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $M: \\frac{x^2}{a^2} + \\frac{y^2}{3} = 1 (a > 0)$ with one of its foci at $F(-1, 0)$. Points $A$ and $B$ are the left and right vertices of the ellipse's major axis, respectively. A line $l$ passes through $F$ and intersects the ellipse at distinct points $C$ ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 13961,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $T$: $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $\\frac{\\sqrt{3}}{2}$, a line passing through the right focus $F$ with slope $k (k > 0)$ intersects $T$ at points $A$ and $B$. If $\\overline{AF} = 3\\overline{FB}$, determine the val... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 13962,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $\\dfrac {x^{2}}{4}+ \\dfrac {y^{2}}{3}=1$ with its left and right foci denoted as $F_{1}$ and $F_{2}$ respectively, and a point $P$ on the ellipse. If $\\overrightarrow{PF_{1}}\\cdot \\overrightarrow{PF_{2}}= \\dfrac {5}{2}$, calculate $| \\overrightarrow{PF_{... | math | {
"style": "rule",
"ground_truth": "\\dfrac{7}{2}"
} | {
"index": 13963,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $\\dfrac {x^{2}}{a^{2}}+ \\dfrac {y^{2}}{b^{2}}=1$ with its left and right foci being $F_{1}$ and $F_{2}$ respectively, a perpendicular line to the x-axis through the left focus $F_{1}(-2,0)$ intersects the ellipse at points $P$ and $Q$. The line $PF_{2}$ inter... | math | {
"style": "rule",
"ground_truth": "- \\dfrac {1}{2}"
} | {
"index": 13964,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $\\dfrac {x^{2}}{a^{2}}+ \\dfrac {y^{2}}{b^{2}}=1(a > b > 0)$, there is a point $P$ on the ellipse such that the distance to the two foci of the ellipse, $F_{1}$ and $F_{2}$, satisfies $|PF_{1}|+|PF_{2}|=10$, and the eccentricity $e= \\dfrac {4}{5}$. \n$(1)$ F... | math | {
"style": "rule",
"ground_truth": "3 \\sqrt {3}"
} | {
"index": 13965,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $\\dfrac{x^2}{a^2} + \\dfrac{y^2}{b^2} = 1 (a > b > 0)$ with its left and right foci being $F_1$ and $F_2$ respectively, and its eccentricity $e = \\dfrac{\\sqrt{2}}{2}$, the length of the minor axis is $2$. \n$(1)$ Find the equation of the ellipse; \n\n$(2)$... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 13966,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $\\frac {x^{2}}{a^{2}} + \\frac {y^{2}}{b^{2}} = 1 (a > b > 0)$ with vertex $B$ at the top, vertex $A$ on the right, and right focus $F$. Let $E$ be a point on the lower half of the ellipse such that the tangent at $E$ is parallel to $AB$. If the eccentricity o... | math | {
"style": "rule",
"ground_truth": "\\frac {\\sqrt {2}}{4}"
} | {
"index": 13967,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $\\frac {x^{2}}{a^{2}} + \\frac {y^{2}}{b^{2}} = 1$ ($a > b > 0$) and a line $l: y = -\\frac { \\sqrt {3}}{3}x + b$ intersect at two distinct points P and Q. The distance from the origin to line $l$ is $\\frac { \\sqrt {3}}{2}$, and the eccentricity of the elli... | math | {
"style": "rule",
"ground_truth": "-\\frac {7}{6}"
} | {
"index": 13968,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $\\frac{x^2}{25} + \\frac{y^2}{9} = 1$, the distance from a point $M$ on the ellipse to the left focus $F_1$ is 8. Find the distance from $M$ to the right directrix."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"index": 13969,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $\\frac{x^2}{25}+\\frac{y^2}{m^2}=1\\left(m \\gt 0\\right)$ with one focus at $F\\left(0,4\\right)$, find $m$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{41}"
} | {
"index": 13970,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $e = \\frac{\\sqrt{6}}{3}$, the distance from the origin to the line passing through points $A(0, -b)$ and $B(a, 0)$ is $\\frac{\\sqrt{3}}{2}$.\n\n$(1)$ Find the equation of the ellipse;\n\... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{6}"
} | {
"index": 13971,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$, its left focus is $F$, left vertex is $A$, and point $B$ is a point on the ellipse in the first quadrant. The line $OB$ intersects the ellipse at another point $C$. If the line $BF$ bisects the line segment... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 13972,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$, let $P$ be a point on the ellipse such that the projection of $P$ onto the $x$-axis is the left focus $F_1$. Let $A$ be the intersection point of the ellipse with the positive semi-major axis, and let $B$ b... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 13973,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ ($a > b > 0$) with a focal distance of $2\\sqrt{3}$, the line $l_1: y = kx$ ($k \\neq 0$) intersects the ellipse at points A and B. A line $l_2$ passing through point B with a slope of $\\frac{1}{4}k$ intersects the ell... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{8}"
} | {
"index": 13974,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ ($a > b > 0$), a line passing through points A($-a, 0$) and B($0, b$) has an inclination angle of $\\frac{\\pi}{6}$, and the distance from origin to this line is $\\frac{\\sqrt{3}}{2}$.\n\n(1) Find the equation of the e... | math | {
"style": "rule",
"ground_truth": "k = \\frac{7}{6}"
} | {
"index": 13975,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ (where $a > b > 0$) with its left focus at F and the eccentricity $e = \\frac{\\sqrt{2}}{2}$, the line segment cut by the ellipse from the line passing through F and perpendicular to the x-axis has length $\\sqrt{2}$.\n... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 13976,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $\\frac{x^{2}}{8} + \\frac{y^{2}}{2} = 1$, and a point $A(2,1)$ on the ellipse. The slopes of the lines $AB$ and $AC$ connecting point $A$ to two moving points $B$ and $C$ on the ellipse are $k_{1}$ and $k_{2}$, respectively, with $k_{1} + k_{2} = 0$. Determine... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 13977,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $\\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity $e = \\frac{\\sqrt{3}}{3}$. The left and right foci are $F_1$ and $F_2$, respectively, with $F_2$ coinciding with the focus of the parabola $y^2 = 4x$.\n(I) Find the standard equ... | math | {
"style": "rule",
"ground_truth": "\\frac{16\\sqrt{3}}{5}"
} | {
"index": 13978,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $\\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1$ ($a>0$, $b>0$) with its left focus $F$ and right vertex $A$, upper vertex $B$. If the distance from point $F$ to line $AB$ is $\\frac{5\\sqrt{14}}{14}b$, find the eccentricity of the ellipse."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 13979,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $\\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1$ (where $a>0$, $b>0$) with eccentricity $\\frac{\\sqrt{6}}{3}$, the distance from the origin O to the line passing through points A $(0, -b)$ and B $(a, 0)$ is $\\frac{\\sqrt{3}}{2}$. Further, the line $y=kx+m$ (... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4}"
} | {
"index": 13980,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1(a>b>0)$ with left and right foci $F_{1}(-c,0)$ and $F_{2}(c,0)$ respectively, and a point $P$ on the ellipse (different from the left and right vertices), the radius of the inscribed circle of $\\triangle PF_{1}F_{2}... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"index": 13981,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse C centered at the origin with its left focus F($-\\sqrt{3}$, 0) and right vertex A(2, 0).\n\n(1) Find the standard equation of ellipse C; \n(2) A line l with a slope of $\\frac{1}{2}$ intersects ellipse C at points A and B. Find the maximum value of the chord ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{10}"
} | {
"index": 13982,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse C: $$\\frac {x^{2}}{a^{2}}$$ + $$\\frac {y^{2}}{b^{2}}$$ = 1 (a > b > 0) with an eccentricity of $$\\frac { \\sqrt {2}}{2}$$, the length of the line segment obtained by intersecting the line y = 1 with the ellipse C is 2$$\\sqrt {2}$$.\n(I) Find the equation of... | math | {
"style": "rule",
"ground_truth": "\\sqrt {6}"
} | {
"index": 13983,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse C: $$\\frac {x^{2}}{a^{2}}+ \\frac {y^{2}}{b^{2}}=1$$ ($a>b>0$) with its left and right foci being $F_1$ and $F_2$ respectively, and a point $$P(1, \\frac {3}{2})$$ on the ellipse such that the line connecting $P$ and the right focus of the ellipse is perpendic... | math | {
"style": "rule",
"ground_truth": "- \\frac { \\sqrt {3}}{12}"
} | {
"index": 13984,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse C: $$\\frac {x^{2}}{a^{2}}+ \\frac {y^{2}}{b^{2}}=1(a＞b＞0)$$ with left and right foci $F_1$ and $F_2$, respectively. Point A is the upper vertex of the ellipse, $|F_{1}A|= \\sqrt {2}$, and the area of △$F_{1}AF_{2}$ is 1.\n(1) Find the standard equation of the ... | math | {
"style": "rule",
"ground_truth": "y=- \\frac {1}{3}"
} | {
"index": 13985,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse C: $$\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$$ (where $a>b>0$) that passes through the point $E(\\sqrt 3, 1)$, with an eccentricity of $\\frac{\\sqrt{6}}{3}$, and O as the origin of the coordinate system.\n(Ⅰ) Find the equation of the ellipse C.\n(Ⅱ) If point P... | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"index": 13986,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse C: $$\\frac{x^2}{a^2}+ \\frac{y^2}{b^2}=1 \\quad (a>b>0)$$ which passes through the point $(1, \\frac{2\\sqrt{3}}{3})$, with its foci denoted as $F_1$ and $F_2$. The circle $x^2+y^2=2$ intersects the line $x+y+b=0$ forming a chord of length 2.\n(I) Determine th... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"index": 13987,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse C: $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ ($a > b > 0$) with an eccentricity of $\\frac{1}{2}$, and a circle centered at the origin with the radius equal to the semi-minor axis of the ellipse is tangent to the line $\\sqrt{7}x - \\sqrt{5}y + 12 = 0$. \n(1) ... | math | {
"style": "rule",
"ground_truth": "-\\frac{12}{7}"
} | {
"index": 13988,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse C: $\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1(a>b>0)$ with its right focus F, upper vertex B, the line BF intersects C at another point A, and the projection of point A on the x-axis is A1. O is the origin, and if $\\overrightarrow{BO}=2\\overrightarrow{A_{1}A}$, det... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"index": 13989,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse C: $\\frac{x^{2}}{3}+y^{2}=1$ with left focus and right focus as $F_{1}$ and $F_{2}$ respectively. The line $y=x+m$ intersects C at points A and B. If the area of $\\triangle F_{1}AB$ is twice the area of $\\triangle F_{2}AB$, find $m$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{2}}{3}"
} | {
"index": 13990,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse E: $$\\frac {x^{2}}{a^{2}}+ \\frac {y^{2}}{b^{2}} = 1$$ ($a > b > 0$) with a focal length of $2\\sqrt{3}$, and the ellipse passes through the point $(\\sqrt{3}, \\frac{1}{2})$.\n(Ⅰ) Find the equation of ellipse E;\n(Ⅱ) Through point P$(-2, 0)$, draw two lines w... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 13991,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse E: $$\\frac {x^{2}}{a^{2}}+ \\frac {y^{2}}{b^{2}}$$\\=1 (a>b>0) passing through point P ($$\\sqrt {3}$$, $$\\frac {1}{2}$$) with its left focus at F ($$-\\sqrt {3}$$, 0).\n\n1. Find the equation of ellipse E.\n2. If A is the right vertex of ellipse E, and the l... | math | {
"style": "rule",
"ground_truth": "$\\frac {2 \\sqrt {5}+ \\sqrt {15}}{4}$"
} | {
"index": 13992,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse E: $$\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$$ ($$a > b > 0$$) passing through point Q ($$\\frac{\\sqrt{2}}{2}, \\frac{\\sqrt{3}}{2}$$), the product of the slopes of the lines connecting the moving point P on the ellipse to the two endpoints of the minor axis i... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 13993,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse M: $$\\frac {x^{2}}{a^{2}}+ \\frac {y^{2}}{b^{2}}=1$$ (a>0, b>0) with two vertices A(-a, 0) and B(a, 0). Point P is a point on the ellipse distinct from A and B. The slopes of lines PA and PB are k₁ and k₂, respectively, and $$k_{1}k_{2}=- \\frac {1}{2}$$.\n(1)... | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {2}}{2}"
} | {
"index": 13994,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse M: $$\\frac {y^{2}}{a^{2}}+ \\frac {x^{2}}{b^{2}}=1$$ (where $a>b>0$) whose eccentricity is the reciprocal of the eccentricity of the hyperbola $x^{2}-y^{2}=1$, and the major axis of the ellipse is 4.\n(1) Find the equation of ellipse M;\n(2) If the line $y= \\... | math | {
"style": "rule",
"ground_truth": "\\sqrt {2}"
} | {
"index": 13995,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse \\( C: \\frac{x^{2}}{2}+y^{2}=1 \\) with left and right foci \\( F_{1}, F_{2} \\) respectively, let \\( P \\) be a point on the ellipse \\( C \\) in the first quadrant. The extended lines \\( P F_{1}, P F_{2} \\) intersect the ellipse \\( C \\) at points \\( Q_... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{2}}{3}"
} | {
"index": 13996,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse \\( C: \\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1 \\) where \\( a > b > 0 \\), with the left focus at \\( F \\). A tangent to the ellipse is drawn at a point \\( A \\) on the ellipse and intersects the \\( y \\)-axis at point \\( Q \\). Let \\( O \\) be the or... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{3}"
} | {
"index": 13997,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse and a hyperbola,\\[\\frac{x^2}{16} - \\frac{y^2}{25} = 1\\]and \\[\\frac{x^2}{K} + \\frac{y^2}{25} = 1\\], have the same asymptotes. Find the value of $K$."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 13998,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse in the Cartesian coordinate system $xOy$, its center is at the origin, the left focus is $F(-\\sqrt{3},0)$, and the right vertex is $D(2,0)$. Let point $A\\left( 1,\\frac{1}{2} \\right)$.\n\n(Ⅰ) Find the standard equation of the ellipse;\n\n(Ⅱ) If a line passin... | math | {
"style": "rule",
"ground_truth": "\\sqrt {2}"
} | {
"index": 13999,
"solution": ""
} |
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