Let the length of the latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(a>b)$, be 30 . If its eccentricity is the maximum value of the function $f(t)=-\frac{3}{4}+2 t-t^2$, then $\left(a^2+b^2\right)$ is equal to

Let each of the two ellipses $\mathrm{E}_1: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(a>b)$ and $\mathrm{E}_2: \frac{x^2}{\mathrm{~A}^2}+\frac{y^2}{\mathrm{~B}^2}=1,(\mathrm{~A}<\mathrm{B})$ have eccentricity $\frac{4}{5}$. Let the lengths of the latus recta of $\mathrm{E}_1$ and $\mathrm{E}_2$ be $l_1$ and $l_2$, respectively, such that $2 l_1^2=9 l_2$. If the distance between the foci of $E_1$ is 8 , then the distance between the foci of $E_2$ is

If the points of intersection of the ellipses $x^2+2 y^2-6 x-12 y+23=0$ and

$4 x^2+2 y^2-20 x-12 y+35=0$ lie on a circle of radius $r$ and centre $(a, b)$, then the

value of $a b+18 r^2$ is :

Let the line $y-x=1$ intersect the ellipse $\frac{x^2}{2}+\frac{y^2}{1}=1$ at the points A and B . Then the angle made by the line segment AB at the center of the ellipse is :

Let S and $\mathrm{S}^{\prime}$ be the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ and $\mathrm{P}(\alpha, \beta)$ be a point on the ellipse in the first quadrant. If $(\mathrm{SP})^2+\left(\mathrm{S}^{\prime} \mathrm{P}\right)^2-\mathrm{SP} \cdot \mathrm{S}^{\prime} \mathrm{P}=37$, then $\alpha^2+\beta^2$ is equal to :

If the line $\alpha x+4 y=\sqrt{7}$, where $\alpha \in \mathbf{R}$, touches the ellipse $3 x^2+4 y^2=1$ at the point P in the first quadrant, then one of the focal distances of $P$ is :

Let $\frac{x^2}{f\left(a^2+7 a+3\right)}+\frac{y^2}{f(3 a+15)}=1$ represent an ellipse with major axis along $y$-axis, where $f$ is a strictly decreasing positive function on $\mathbf{R}$. If the set of all possible values of $a$ is $\mathbf{R}-[\alpha, \beta]$, then $\alpha^2+\beta^2$ is equal to :

Let $x=9$ be a directrix of an ellipse E , whose centre is at the origin and eccentricity is $\frac{1}{3}$. Let $\mathrm{P}(\alpha, 0)$, $\alpha>0$, be a focus of E and AB be a chord passing through P . Then the locus of the mid point of AB is :

Let a focus of the ellipse $\mathrm{E}: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ be $\mathrm{S}(4,0)$ and its eccentricity be $\frac{4}{5}$. If the point $\mathrm{P}(3, \alpha)$ lies on E and O is the origin, then the area of $\triangle \mathrm{POS}$ is equal to:

Let $\mathrm{P}(3 \cos \alpha, 2 \sin \alpha), \alpha \neq 0$, be a point on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1, \mathrm{Q}$ be a point on the circle $x^2+y^2-14 x-14 y+82=0$ and R be a point on the line $x+y=5$ such that the centroid of the triangle PQR is $\left(2+\cos \alpha, 3+\frac{2}{3} \sin \alpha\right)$. Then the sum of the ordinates of all possible points R is:

Let an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, $a < b$, pass through the point (4, 3) and have eccentricity $\frac{\sqrt{5}}{3}$.

Then the length of its latus rectum is :

Let the ellipse $3x^2 + py^2 = 4$ pass through the centre $C$ of the circle $x^2 + y^2 - 2x - 4y - 11 = 0$ of radius $r$. Let $f_1, f_2$ be the focal distances of the point $C$ on the ellipse. Then $6f_1f_2 - r$ is equal to

Let the length of a latus rectum of an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ be 10. If its eccentricity is the minimum value of the function $f(t) = t^2 + t + \frac{11}{12}$, $t \in \mathbb{R}$, then $a^2 + b^2$ is equal to :

Let p be the number of all triangles that can be formed by joining the vertices of a regular polygon P of n sides and q be the number of all quadrilaterals that can be formed by joining the vertices of P. If p + q = 126, then the eccentricity of the ellipse $\frac{x^2}{16} + \frac{y^2}{n} = 1$ is :

Let for two distinct values of p the lines $y=x+\mathrm{p}$ touch the ellipse $\mathrm{E}: \frac{x^2}{4^2}+\frac{y^2}{3^2}=1$ at the points A and B . Let the line $y=x$ intersect E at the points C and D . Then the area of the quadrilateral $A B C D$ is equal to :

The centre of a circle C is at the centre of the ellipse $\mathrm{E}: \frac{x^2}{\mathrm{a}^2}+\frac{y^2}{\mathrm{~b}^2}=1, \mathrm{a}>\mathrm{b}$. Let C pass through the foci $F_1$ and $F_2$ of E such that the circle $C$ and the ellipse $E$ intersect at four points. Let P be one of these four points. If the area of the triangle $\mathrm{PF}_1 \mathrm{~F}_2$ is 30 and the length of the major axis of $E$ is 17 , then the distance between the foci of $E$ is :

The length of the latus-rectum of the ellipse, whose foci are $(2,5)$ and $(2,-3)$ and eccentricity is $\frac{4}{5}$, is

A line passing through the point $P(\sqrt{5}, \sqrt{5})$ intersects the ellipse $\frac{x^2}{36}+\frac{y^2}{25}=1$ at $A$ and $B$ such that $(P A) \cdot(P B)$ is maximum. Then $5\left(P A^2+P B^2\right)$ is equal to :

If $S$ and $S^{\prime}$ are the foci of the ellipse $\frac{x^2}{18}+\frac{y^2}{9}=1$ and P be a point on the ellipse, then $\min \left(S P \cdot S^{\prime} P\right)+\max \left(S P \cdot S^{\prime} P\right)$ is equal to :

Let the ellipse $E_1: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, $a > b$ and $E_2: \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$, $A < B$ have same eccentricity $\frac{1}{\sqrt{3}}$. Let the product of their lengths of latus rectums be $\frac{32}{\sqrt{3}}$ and the distance between the foci of $E_1$ be 4. If $E_1$ and $E_2$ meet at A, B, C and D, then the area of the quadrilateral ABCD equals :

The equation of the chord, of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$, whose mid-point is $(3,1)$ is :

Let the product of the focal distances of the point $\left(\sqrt{3}, \frac{1}{2}\right)$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(a>b)$, be $\frac{7}{4}$. Then the absolute difference of the eccentricities of two such ellipses is

The length of the chord of the ellipse $\frac{x^2}{4}+\frac{y^2}{2}=1$, whose mid-point is $\left(1, \frac{1}{2}\right)$, is :

Let $\mathrm{E}: \frac{x^2}{\mathrm{a}^2}+\frac{y^2}{\mathrm{~b}^2}=1, \mathrm{a}>\mathrm{b}$ and $\mathrm{H}: \frac{x^2}{\mathrm{~A}^2}-\frac{y^2}{\mathrm{~B}^2}=1$. Let the distance between the foci of E and the foci of $H$ be $2 \sqrt{3}$. If $a-A=2$, and the ratio of the eccentricities of $E$ and $H$ is $\frac{1}{3}$, then the sum of the lengths of their latus rectums is equal to :

Let $f(x)=x^2+9, g(x)=\frac{x}{x-9}$ and $\mathrm{a}=f \circ g(10), \mathrm{b}=g \circ f(3)$. If $\mathrm{e}$ and $l$ denote the eccentricity and the length of the latus rectum of the ellipse $\frac{x^2}{\mathrm{a}}+\frac{y^2}{\mathrm{~b}}=1$, then $8 \mathrm{e}^2+l^2$ is equal to.

Let the line $2 x+3 y-\mathrm{k}=0, \mathrm{k}>0$, intersect the $x$-axis and $y$-axis at the points $\mathrm{A}$ and $\mathrm{B}$, respectively. If the equation of the circle having the line segment $A B$ as a diameter is $x^2+y^2-3 x-2 y=0$ and the length of the latus rectum of the ellipse $x^2+9 y^2=k^2$ is $\frac{m}{n}$, where $m$ and $n$ are coprime, then $2 \mathrm{~m}+\mathrm{n}$ is equal to

Let $P$ be a parabola with vertex $(2,3)$ and directrix $2 x+y=6$. Let an ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b$, of eccentricity $\frac{1}{\sqrt{2}}$ pass through the focus of the parabola $P$. Then, the square of the length of the latus rectum of $E$, is

Let $A(\alpha, 0)$ and $B(0, \beta)$ be the points on the line $5 x+7 y=50$. Let the point $P$ divide the line segment $A B$ internally in the ratio $7:3$. Let $3 x-25=0$ be a directrix of the ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and the corresponding focus be $S$. If from $S$, the perpendicular on the $x$-axis passes through $P$, then the length of the latus rectum of $E$ is equal to,

If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :

Let the tangent and normal at the point $(3 \sqrt{3}, 1)$ on the ellipse $\frac{x^{2}}{36}+\frac{y^{2}}{4}=1$ meet the $y$-axis at the points $A$ and $B$ respectively. Let the circle $C$ be drawn taking $A B$ as a diameter and the line $x=2 \sqrt{5}$ intersect $C$ at the points $P$ and $Q$. If the tangents at the points $P$ and $Q$ on the circle intersect at the point $(\alpha, \beta)$, then $\alpha^{2}-\beta^{2}$ is equal to :

Let $\mathrm{P}\left(\frac{2 \sqrt{3}}{\sqrt{7}}, \frac{6}{\sqrt{7}}\right), \mathrm{Q}, \mathrm{R}$ and $\mathrm{S}$ be four points on the ellipse $9 x^{2}+4 y^{2}=36$. Let $\mathrm{PQ}$ and $\mathrm{RS}$ be mutually perpendicular and pass through the origin. If $\frac{1}{(P Q)^{2}}+\frac{1}{(R S)^{2}}=\frac{p}{q}$, where $p$ and $q$ are coprime, then $p+q$ is equal to :

If the radius of the largest circle with centre (2,0) inscribed in the ellipse $x^2+4y^2=36$ is r, then 12r$^2$ is equal to :

Consider ellipses $\mathrm{E}_{k}: k x^{2}+k^{2} y^{2}=1, k=1,2, \ldots, 20$. Let $\mathrm{C}_{k}$ be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse $\mathrm{E}_{k}$. If $r_{k}$ is the radius of the circle $\mathrm{C}_{k}$, then the value of $\sum_\limits{k=1}^{20} \frac{1}{r_{k}^{2}}$ is :

Let a circle of radius 4 be concentric to the ellipse $15 x^{2}+19 y^{2}=285$. Then the common tangents are inclined to the minor axis of the ellipse at the angle :

Let the ellipse $E:{x^2} + 9{y^2} = 9$ intersect the positive x and y-axes at the points A and B respectively. Let the major axis of E be a diameter of the circle C. Let the line passing through A and B meet the circle C at the point P. If the area of the triangle with vertices A, P and the origin O is ${m \over n}$, where m and n are coprime, then $m - n$ is equal to :

In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of persons, who speak only English is $\alpha$ and the number of persons who speak only Hindi is $\beta$, then the eccentricity of the ellipse $25\left(\beta^{2} x^{2}+\alpha^{2} y^{2}\right)=\alpha^{2} \beta^{2}$ is :

If the maximum distance of normal to the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{b^{2}}=1, b < 2$, from the origin is 1, then the eccentricity of the ellipse is :

Let a line L pass through the point of intersection of the lines $b x+10 y-8=0$ and $2 x-3 y=0, \mathrm{~b} \in \mathbf{R}-\left\{\frac{4}{3}\right\}$. If the line $\mathrm{L}$ also passes through the point $(1,1)$ and touches the circle $17\left(x^{2}+y^{2}\right)=16$, then the eccentricity of the ellipse $\frac{x^{2}}{5}+\frac{y^{2}}{\mathrm{~b}^{2}}=1$ is :

The acute angle between the pair of tangents drawn to the ellipse $2 x^{2}+3 y^{2}=5$ from the point $(1,3)$ is :

If the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ meets the line $\frac{x}{7}+\frac{y}{2 \sqrt{6}}=1$ on the $x$-axis and the line $\frac{x}{7}-\frac{y}{2 \sqrt{6}}=1$ on the $y$-axis, then the eccentricity of the ellipse is :

Let the eccentricity of the ellipse ${x^2} + {a^2}{y^2} = 25{a^2}$ be b times the eccentricity of the hyperbola ${x^2} - {a^2}{y^2} = 5$, where a is the minimum distance between the curves y = e^x and y = log_ex. Then ${a^2} + {1 \over {{b^2}}}$ is equal to :

Let the eccentricity of an ellipse ${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$, $a > b$, be ${1 \over 4}$. If this ellipse passes through the point $\left( { - 4\sqrt {{2 \over 5}} ,3} \right)$, then ${a^2} + {b^2}$ is equal to :

If m is the slope of a common tangent to the curves ${{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1$ and ${x^2} + {y^2} = 12$, then $12{m^2}$ is equal to :

The locus of the mid point of the line segment joining the point (4, 3) and the points on the ellipse ${x^2} + 2{y^2} = 4$ is an ellipse with eccentricity :

The line y = x + 1 meets the ellipse ${{{x^2}} \over 4} + {{{y^2}} \over 2} = 1$ at two points P and Q. If r is the radius of the circle with PQ as diameter then (3r)² is equal to :

Let the maximum area of the triangle that can be inscribed in the ellipse ${{{x^2}} \over {{a^2}}} + {{{y^2}} \over 4} = 1,\,a > 2$, having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the y-axis, be $6\sqrt 3 $. Then the eccentricity of the ellipse is :

STATEMENT-2 :If line $y = mx + {{4\sqrt 3 } \over m},\left( {m \ne 0} \right)$ is a common tangent to the parabola ${y^2} = 16\sqrt {3x} $and the ellipse $2{x^2} + {y^2} = 4$, then $m$ satisfies ${m^4} + 2{m^2} = 24$