Embibe Experts Solutions for Chapter: Hyperbola, Exercise 3: Level 3

If two points $P$ and $Q$ on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,$ whose centre $C$ be such that $CP$ is perpendicular to $CQ, a<b,$  then the value of $\frac{1}{C{P}^2}+\frac{1}{C{Q}^2}$ is

If a chord joining $P\left(a\sec \theta , a\tan \theta \right)$ and $Q\left(a\sec \alpha , a\tan \alpha \right)$ on the hyperbola $x^2-y^2=a^2$ is the normal at $P$, then $\tan \alpha$ is

If $PQ$ is a double ordinate of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ such that $OPQ$ is an equilateral triangle, $O$ being the centre of the hyperbola, then the eccentricity $e$ of the hyperbola satisfies

The foci of a hyperbola coincide with the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$. If the eccentricity of the hyperbola is $2$, then the equation of the tangent to this hyperbola passing through the point $\left(4,6\right)$ is

A square $ABCD$ has all its vertices on the curve $x^2{y}^2=1.$ The midpoints of its sides also lie on the same curve. Then, the square of area of $ABCD$ is