Amit M Agarwal Solutions for Chapter: Ellipse, Exercise 5: Entrances Gallery

$S$ and $T$ are the foci of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $B$ is the end of the minor axis. If  $\mathrm{\Delta }STB$ is an equilateral triangle, then eccentricity of the ellipse is

The line $x=2y$ intersects the ellipse $\frac{x^2}{4}+y^2=1$ at the points $P$ and $Q.$ The equation of the circle with $PQ$ as diameter is

If the foci of the ellipse $\frac{x^2}{9}+y^2=1$ subtend a right angle at a point $P$. Then, the locus of $P$ is

If the centre, one of the foci and semi-major axis of an ellipse are $\left(0,0\right),\left(0,3\right)$ and $5,$ then its equation is

The length of the latus rectum of the ellipse $16x^2+25y^2=400$ is

If a point $P\left(x,y\right)$ moves along the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and $C$ is the centre of the ellipse, then the sum of maximum and minimum values of $CP$ is

If the length of the major axis of an ellipse is $\frac{17}{8}$ times the length of the minor axis, then the eccentricity of the ellipse is

Area of a triangle formed by tangent and normal to the curve $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at $P\left(\frac{a}{\sqrt{2}},\frac{b}{\sqrt{2}}\right)$ with the $X$ -axis is