Basics of Ellipse

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

Let an ellipse with centre $\left(1, 0\right)$ and latus rectum of length $\frac{1}{2}$ have its major axis along x-axis. If its minor axis subtends an angle ${60}^{\circ }$ at the foci, then the square of the sum of the lengths of its minor and major axes is equal to _______.

Consider ellipses $E_k:k{x}^2+k^2{y}^2=1,k=1,2,\dots ,20$. Let $C_k$ be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse $E_k$. If $r_k$ is the radius of the circle $C_k$, then the value of $\sum _{k=1}^{20}\frac{1}{r_k^2}$ is

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 speaks 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

The length of the latus rectum of the ellipse $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ is $16$, then the number of possible value$\left(s\right)$ of $b$ is

If the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is inscribed in the rectangle whose length to breadth are in the ratio $2:1$, then the area of the rectangle is

A circle concentric to the ellipse $\frac{4x^2}{289}+\frac{y^2}{{\lambda }^2}=1\left(\lambda <\frac{17}{2}\right)$ pass through foci $F_1 \& F_2$ and cuts the ellipse at point $P$. If area of $∆P{F}_1{F}_2$ is $30$ sq. units, then $\frac{F_1{F}_2}{13}=$

${\left(x-2\right)}^2+{\left(y+3\right)}^2=16$ touching the ellipse $\frac{{\left(x-2\right)}^2}{p^2}+\frac{{\left(y+3\right)}^2}{q^2}=1$ from inside. If $\left(2,-6\right)$ is one focus of ellipse, then $\left(p,q\right)$ is

If $e_1 \& e_2$ are the eccentricities of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$ respectively, then

If point $P$ lies outside of circle $x^2+y^2-2x=0$ and moves such that its distance from nearest point on the circle is half of its distance from line $x=5$, then locus of point $P$ can be

If $\frac{x^2}{3-\lambda }+\frac{y^2}{\lambda -5}+1=0$ represent an ellipse whose major axis lies along $y-$axis then

Let the two concentric ellipses be such that foci of one be on the other and the angle between their principal axes is ${\cos }^{-1}\frac{\sqrt{5}}{3}$. If the eccentricity of one ellipse is $\frac{2}{\sqrt{5}}$, then the eccentricity of other ellipse is

If the point $M$ lies on the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$, then the maximum distance between $M \& N\equiv \left(6,0\right)$ is

Find the length of the focal chord of the ellipse $\frac{x^2}{4^2}+\frac{y^2}{3^2}=1$, which is inclined to the major axis at angle $60°$.

Find the length of the focal chord of the ellipse $\frac{x^2}{3^2}+\frac{y^2}{2^2}=1$, which is inclined to the major axis at angle $60°$.

Find the length of the focal chord of the ellipse $\frac{x^2}{3^2}+\frac{y^2}{2^2}=1$, which is inclined to the major axis at angle $30°$.

Find the length of the focal chord of the ellipse $\frac{x^2}{3^2}+\frac{y^2}{2^2}=1$, which is inclined to the major axis at angle $45°$.

If an ellipse has its foci at $\left(2,0\right)$ and $\left(-2,0\right)$ and its length of the latus rectum is $6$, then the equation of the ellipse is

The ratio of the area enclosed by the locus of the midpoint of $PS$ and area of the ellipse is $\frac{1}{\alpha }$ then the value of $10\alpha$ is ($P$ be any point on the ellipse and $S$, its focus)

A chord is drawn passing through $P\left(2, 2\right)$ on the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ such that it intersects the ellipse at points $A$ and $B$. Then the maximum value of $PA·PB$ is equal to $\frac{a}{b}$ where $a,b$ are the least common multiple then the value of $a+b$ is