Basics of Ellipse

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 $\frac{1}{2}\left(a^2{\beta }^2+b^2{\alpha }^2\right)=$

The equation of an ellipse whose focus is $\left(-1,\mathrm{ }1\right)$ , whose directrix is $x-y+3=0$ and whose eccentricity is $\frac{1}{2}$ , is given by

The equation of the circle passing through the focii of the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$, and having centre at $\left(0,3\right)$ is

The sum of the focal distances of any point on the conic $\frac{x^2}{25}+\frac{y^2}{16}=1$ 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 $PA+PB=$

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 ${\cos }^{-1}\frac{\sqrt{e^2+e^{\text{'}2}-1}}{e{e}^{\text{'}}}$.

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 co-ordinates of foci of the ellipse $16x^2+9y^2=144$.

Show that the point $(2, -3)$ lies inside the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$.

Show that the point $x=\frac{a\left(1-t^2\right)}{1+t^2}, y=\frac{2bt}{1+t^2}$, when $t$ is a parameter lies on an ellipse.

Prove that the locus of the point of intersection of $\frac{tx}{a}+\frac{y}{b}-t=0$ and $\frac{x}{a}-\frac{ty}{b}+1=0$ (where $t$ is a parameter) is an ellipse.

The coordinates of a focus of an ellipse are $(0, -3)$, the equation of its corresponding directrix is $3x-4y+1=0$ and its eccentricity is $\frac{1}{\sqrt{2}}$. Find the equation of the ellipse.

Prove that the equation $52\left(x^2+y^2\right)=(2x+3y+1{)}^2$ represents an ellipse. Find the eccentricity of the ellipse.

If co-ordinates of a variable point $P$ are $\left(4\frac{1-t^2}{1+t^2}, \frac{6t}{1+t^2}\right)$, where $t$ is a variable, then prove that the locus of $P$ is an ellipse whose centre is at the origin and the co-ordinates of the vertices are $(\pm 4, 0)$.

Determine the position of the following points with respect to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$.

[i] $(1,1)$, [ii] $\left(4,3\right)$, [iii] $\left(\frac{3\sqrt{3}}{2},1\right)$

Find the equation of locus of a variable point whose parametric co-ordinates are $(2+3 \cos \theta , 3+2 \sin \theta )$.

Find the equation of locus of a variable point $P$ whose co-ordinates are $(4 \cos \theta , 3 \sin \theta )$.

The eccentricity of the ellipse $\frac{{\left(x - 1\right)}^2}{2}+{\left(y+\frac{3}{4}\right)}^2=\frac{1}{16}$ is

The coordinates of points $A$ and $B$ shown on a ellipse, whose equation is given by $4x^2+9y^2=36,$ are :-

Find the condition for the line $\mathrm{xcos\alpha }+\mathrm{ysin\alpha }=\mathrm{p}$ to be a tangent to the ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$