If $\alpha x+\beta y=109$ is the equation of the chord of the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$, whose mid point is $\left(\frac{5}{2},\frac{1}{2}\right)$. then $\alpha +\beta$ is equal to :

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 :

If the midpoint of a chord of the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ is $(\sqrt{2},4/3)$, and the length of the chord is $\frac{2\sqrt{\alpha }}{3}$, then $\alpha$ 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 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 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 ellipse $3x^2+p{y}^2=4$ 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 $6f_1{f}_2-r$ 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 $\min \left(SP\cdot S^{\prime }P\right)+\max \left(SP\cdot S^{\prime }P\right)$ 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 :

If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse 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 $(PA)\cdot (PB)$ is maximum. Then $5\left(P{A}^2+P{B}^2\right)$ is equal to :

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

Let $C$ be the circle of minimum area enclosing the ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with eccentricity $\frac{1}{2}$ and foci $(\pm 2,0)$. Let $P Q R$ be a variable triangle, whose vertex $P$ is on the circle $C$ and the side $Q R$ of length $2 a$ is parallel to the major axis of $E$ and contains the point of intersection of $E$ with the negative $y$-axis. Then the maximum area of the triangle $P Q R$ is :

Let $\mathrm{S}=\mathbf{N}\cup \{0\}$. Define a relation R from S to $\mathbf{R}$ by : $$\mathrm{R}=\left\{(x,y):{\log }_{\mathrm{e}}y=x{\log }_{\mathrm{e}}\left(\frac{2}{5}\right),x\in \mathrm{S},y\in \mathbf{R}\right\}.$$ Then, the sum of all the elements in the range of $R$ is equal to :

Define a relation R on the interval $\left[0,\frac{\pi }{2}\right)$ by $ x $ R $ y $ if and only if ${\sec }^{2}x-{\tan }^{2}y=1$. Then R is :

Let $A=\{1,2,3,\dots ,10\}$ and $B=\left\{\frac{m}{n}: m, n \in A, m

The number of non-empty equivalence relations on the set $\{1,2,3\}$ is :