Latus rectum of the conic satisfying the differential equation, $ x dy + y dx = 0$  and passing through the point $ (2, 8) $ is :

The product of the lengths of perpendiculars drawn from any point on the hyperbola $x^2 -2y^2 -2=0$  to its asymptotes is 

Let the foci of a hyperbola $\mathrm{H}$ coincide with the foci of the ellipse $E: \frac{(x-1)^2}{100}+\frac{(y-1)^2}{75}=1$ and the eccentricity of the hyperbola $\mathrm{H}$ be the reciprocal of the eccentricity of the ellipse $E$. If the length of the transverse axis of $\mathrm{H}$ is $\alpha$ and the length of its conjugate axis is $\beta$, then $3 \alpha^2+2 \beta^2$ is equal to :

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

If ${m_1}$ and ${m_2}$are the slopes of the tangents to the hyperbola $\frac{{{x^2}}}{{25}} – \frac{{{y^2}}}{{16}} = 1$ which pass through the point $(6, 2)$, then