The tangent to the hyperbola, $x^2 - 3y^2 = 3$  at the point $\left( {\sqrt 3 \,\,,\,\,0} \right)$ when associated with two asymptotes constitutes :

The number of possible tangents which can be drawn to the curve $4x^2 - 9y^2 = 36$ , which are perpendicular to the straight line $5x + 2y -10 = 0$ is

The foci of a hyperbola are $( \pm 2,0)$ and its eccentricity is $\frac{3}{2}$. A tangent, perpendicular to the line $2 x+3 y=6$, is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the $x$ - and $y$-axes are $a$ and $b$ respectively, then $|6 a|+|5 b|$ is equal to $..........$.

Let tangents drawn from point $C(0,-b)$ to hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ touches hyperbola at points $A$ and $B.$ If $\Delta ABC$ is a right angled triangle, then $\frac{a^2}{b^2}$ is equal to -

Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola $\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$

An ellipse $E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ passes through the vertices of the hyperbola $H: \frac{x^{2}}{49}-\frac{y^{2}}{64}=-1$. Let the major and minor axes of the ellipse $E$ coincide with the transverse and conjugate axes of the hyperbola $H$. Let the product of the eccentricities of $E$ and $H$ be $\frac{1}{2}$. If $l$ is the length of the latus rectum of the ellipse $E$, then the value of $113\,l$ is equal to $....$