Let $P (3\, sec\,\theta , 2\, tan\,\theta )$ and $Q\, (3\, sec\,\phi , 2\, tan\,\phi )$ where $\theta + \phi \, = \frac{\pi}{2}$ , be two distinct points on the hyperbola $\frac{{{x^2}}}{9} – \frac{{{y^2}}}{4} = 1$ . Then the ordinate of the point of intersection of the normals at $P$ and $Q$ is

Let $a>0, b>0$. Let $e$ and $\ell$ respectively be the eccentricity and length of the latus rectum of the hyperbola $\frac{ x ^{2}}{ a ^{2}}-\frac{ y ^{2}}{ b ^{2}}=1$. Let $e ^{\prime}$ and $\ell^{\prime}$ respectively the eccentricity and length of the latus rectum of its conjugate hyperbola. If $e ^{2}=\frac{11}{14} \ell$ and $\left( e ^{\prime}\right)^{2}=\frac{11}{8} \ell^{\prime}$, then the value of $77 a+44 b$ is equal to

The equation of the hyperbola referred to its axes as axes of coordinate and whose distance between the foci is $16$ and eccentricity is $\sqrt 2 $, is

The equation of the tangent to the conic ${x^2} – {y^2} – 8x + 2y + 11 = 0$ at $(2, 1)$ is

The asymptote of the hyperbola $\frac{{{x^2}}}{{{a^2}}} – \frac{{{y^2}}}{{{b^2}}}= 1$ form with any tangent to the hyperbola a triangle whose area is $a^2$ $\tan$ $ \lambda $ in magnitude then its eccentricity is :