The locus of a point $P (h, k)$ such that the line $y = hx + k$ is tangent to $4x^2 - 3y^2 = 1$ , is a/an

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

A point on the curve $\frac{{{x^2}}}{{{A^2}}} - \frac{{{y^2}}}{{{B^2}}} = 1$ 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 eccentricity of the hyperbola ${x^2} - {y^2} = 25$ is

The latus-rectum of the hyperbola $16{x^2} - 9{y^2} = $ $144$, is