Basics of 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

Let $\alpha$  and $f\left(\alpha \right)$ be the eccentricity of the ellipse $\frac{x^2}{4b^2-3a^2}+\frac{y^2}{3b^2-2a^2}=1$; $\left(b^2>a^2\right)$ and $\frac{x^2}{3b^2-2a^2}+\frac{y^2}{b^2}=1$; $\left(b^2>a^2\right)$ respectively, then

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 

The equation $\sqrt{x^2+{\left(y-2\right)}^2}+\sqrt{x^2+{\left(y-4\right)}^2}=k$ will represent

Consider two concentric circles $C_1:x^2+y^2=1$ and $C_2:x^2+y^2=4$. A parabola is drawn through the points where $C_1$ meets $x$-axis and having arbitrary tangent of $C_2$ as its directrix. The focus of the parabola always lies on

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 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 distance between the directrices of the ellipse ${\left(4x-8\right)}^2+16y^2={\left(x+\sqrt{3}y+10\right)}^2$ is

If $A$ and $B$ are foci of ellipse ${\left(x-2y+3\right)}^2+{\left(8x+4y+4\right)}^2=20$ and $P$ is any point on it, then $PA+PB=$

In the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, the lines joining the ends of the minor axis to one of the focus are at right angles. Also, the distance between the focus and the nearer vertex is $\sqrt{10}-\sqrt{5}$. Then the value of $\sqrt{a^2+b^2+1}$ is

Let $P$ and $P^{\text{'}}$ be the extremities of a focal chord of the ellipse $9x^2+16y^2=144$ one of whose foci is at $S$. Show that $\frac{1}{SP}+\frac{1}{S{P}^{\text{'}}}=\frac{8}{9}$.

If two concentric ellipses be such that the foci of one lie on the other and if $e$ and $e^{\text{'}}$ be their eccentricities, show that their axes are inclined at an angle ${\cos }^{-1}\frac{\sqrt{e^2+e^{\text{'}2}-1}}{e{e}^{\text{'}}}$.

Find the length of the chord intercepeted by the ellipse $4x^2+y^2=16$ on the line $y=2x-1$.

Find the eccentricity of the ellipse which meets the straight line $\frac{x}{7}+\frac{y}{2}=1$ on the $x$-axis and the straight line $\frac{x}{3}-\frac{y}{5}=1$ on the $y$-axis, and whose axes are along the axes of co-ordinates.

A chord of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ subtends a right angle at $(a, 0)$. The eccentric angles of the end points of the chord are $\alpha$ and $\beta$; Prove that $\tan \frac{\alpha }{2}\tan \frac{\beta }{2}=\frac{-b^2}{a^2}$.

Find the eccentricity of the curve represented by $x^2+2y^2-2x+3y+2=0$.

Find the length of latus rectum of the ellipse $5x^2+9y^2=45$.

Find the eccentricity of the ellipse, if the minor axis is equal to the distance between the foci.

Find the eccentricity of the ellipse, if the distance between the foci is equal to the length of latus rectum.

Find the eccentricity of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ if its latus rectum is equal to one-half of its minor axis.