Let $\lambda x-2 y=\mu$ be a tangent to the hyperbola $a^{2} x^{2}-y^{2}=b^{2}$. Then $\left(\frac{\lambda}{a}\right)^{2}-\left(\frac{\mu}{b}\right)^{2}$ is equal to

Let $\mathrm{A}\,(\sec \theta, 2 \tan \theta)$ and $\mathrm{B}\,(\sec \phi, 2 \tan \phi)$, where $\theta+\phi=\pi / 2$, be two points on the hyperbola $2 \mathrm{x}^{2}-\mathrm{y}^{2}=2$. If $(\alpha, \beta)$ is the point of the intersection of the normals to the hyperbola at $\mathrm{A}$ and $\mathrm{B}$, then $(2 \beta)^{2}$ is equal to ..... .

Let the hyperbola $H : \frac{ x ^{2}}{ a ^{2}}- y ^{2}=1$ and the ellipse $E: 3 x^{2}+4 y^{2}=12$ be such that the length of latus rectum of $H$ is equal to the length of latus rectum of $E$. If $e_{ H }$ and $e_{ E }$ are the eccentricities of $H$ and $E$ respectively, then the value of $12\left( e _{ H }^{2}+ e _{ E }^{2}\right)$ is equal to

The equation of the normal at the point $(a\sec \theta ,\;b\tan \theta )$ of the curve ${b^2}{x^2} - {a^2}{y^2} = {a^2}{b^2}$ is

The equation of the hyperbola referred to the axis as axes of co-ordinate and whose distance between the foci is $16$ and eccentricity is $\sqrt 2 $, is

A tangent to a hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ intercepts a length of unity from each of the co-ordinate axes, then the point $(a, b)$ lies on the rectangular hyperbola