Embibe Experts Solutions for Chapter: Hyperbola, Exercise 2: Level 2

If the distance between the foci and the distance between the two directrix of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ are in the ratio $3:2,$ then $b:a$ is

If the eccentricity of the hyperbola $x^2-y^2{\sec }^2\alpha =5$ is $\sqrt{3}$  times the eccentricity of the ellipse $x^2{\sec }^2\alpha +y^2=25,$ then a value of $\alpha$ is

The locus of the point of intersection of the lines $\left(\sqrt{3}\right)kx+ky-4\sqrt{3}=0$ and $\sqrt{3}x-y-4\left(\sqrt{3}\right)k=0$ is a conic, whose eccentricity is

A tangent drawn to hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ at $P\left(\frac{\pi }{6}\right)$ forms a triangle of area $3a^2 \mathrm{sq}.\mathrm{units}$, with coordinate axes, then the square of its eccentricity is

The locus of a perpendicular from the center upon the normal to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is

The locus of the foot of the perpendicular from the centre of the hyperbola $xy=1$ on a variable tangent 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