Ellipse

$P\left(x, y\right)$ is any point on the ellipse $x^2+4y^2=1$. Let $E, F$ be the foci of the ellipse.

Consider the following points :

$1.$ $\left(\frac{\sqrt{3}}{2}, 0\right)$

$2.$ $\left(\frac{\sqrt{3}}{2}, \frac{1}{4}\right)$

$3.$ $\left(\frac{\sqrt{3}}{2}, -\frac{1}{4}\right)$

Which of the above points lie on latus rectum of ellipse?

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

If the eccentricity of an ellipse is $\frac{5}{8}$ and distance between its foci is $6 \mathrm{units}$, then the length of latus rectum of the ellipse is $\frac{117}{k}$ where $k$ is 

If a triangle is inscribed in an ellipse and two of its sides are parallel to the given straight lines, then prove that the third side touches the fixed ellipse.

If $S$ and $S^{\text{'}}$ are the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and $P$ is any point on it, then range of values of $SP·S^{\text{'}}P$ is

If there is exactly one tangent at a distance of $4$ units from one focus $\frac{x^2}{a^2}+\frac{y^2}{a^2-16}=1$; $a>4$, then length of latus rectum 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

$(2,6) and (12,4)$ are focii of an ellipse which touches X axis. Find the equation of ellipse

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 $\frac{F_1{F}_2}{13}=$

Aastha is sketching the curves ${\mathrm{P}}_1:{\mathrm{y}}^2=4\mathrm{ax},{\mathrm{P}}_2:{\mathrm{y}}^2=-4\mathrm{ax}, \mathrm{L}:\mathrm{y}=\mathrm{x}$

If $\mathrm{a}=4$, then equation of the ellipse having the line joining the focii of the parabolas ${\mathrm{P}}_1 \mathrm{and} {\mathrm{P}}_2$​ as the major axis and eccentricity equal to $\frac{1}{2}$​ is

${\left(x-2\right)}^2+{\left(y+3\right)}^2=16$ 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

Given points $P,Q$ on an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \left(a>b>0\right)$ satisfy $OP\perp OQ$, the minimum of $\left|OP\right|\times \left|OQ\right|$ is 

$P$ and $Q$ are two points of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$,such that sum of their ordinates is $3.$ Prove that the locus of the intersection of the tangents at $P$ and $Q$ is $9x^2+25y^2=150y.$

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 the eccentricity and the length of the latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are $\frac{\sqrt{3}}{2}$ and $1$ respectively, then the sum of the lengths of major axis and minor axis of the ellipse is

$P\left({\theta }_1\right)$ and $Q\left({\theta }_2\right)$ are two points on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with eccentricity $e$. If $PSQ$ is a focal chord and $\tan \left(\frac{{\theta }_1}{2}\right)\tan \left(\frac{{\theta }_2}{2}\right)=-\left(2\sqrt{2}+3\right)$, then $e$ and $S$ are

Let $S\equiv \frac{x^2}{a^2}+\frac{y^2}{b^2}-1=0,S^{\text{'}}\equiv \frac{x^2}{{\alpha }^2}+\frac{y^2}{{\beta }^2}-1=0$ be two intersecting ellipses. If $P\left(a\cos \theta ,b\sin \theta \right)$ and $Q\left(a\cos \left(\frac{\pi }{2}+\theta \right),b\sin \left(\frac{\pi }{2}+\theta \right)\right)$ are their points of intersection then $\frac{1}{2}\left(a^2{\beta }^2+b^2{\alpha }^2\right)=$