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

The value of $m$, for which the line $y=mx+2$ is a tangent to the hyperbola $4x^2-9y^2=36$, are

The eccentricity of the hyperbola with latus rectum $12$ and semi-conjugate axis $2\sqrt{3}$, is

The equation to the common tangents to the two hyperbolas $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$ are

The equation of the common tangent to the parabola $y^2=8x$ and rectangular hyperbola $xy=-1$ is

The equation of a hyperbola, whose asymptotes are $3x\pm 5y=0$ and vertices are $\left(\pm 5,0\right)$

Equation of auxiliary circle of $\frac{x^2}{16}-\frac{y^2}{25}=1$, is

The distance between the foci of the conic $7x^2-9y^2=63$ is equal to

For different values of $\alpha$, the locus of the point of intersection of the two straight lines $\sqrt{3}x-y-4\sqrt{3}\alpha =0$ and $\sqrt{3}\alpha x+\alpha y-4\sqrt{3}=0$ is