Let $P(h, k)$ be the point of contact of the tangent to the hyperbola $5 x^2-7 y^2-35=0$ which is parallel to the line $\sqrt{2} x-y+\lambda=0$. If $P$ lies in the third quadrant,then $3 h^2-2 k=$

Let $H_1: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $H_2:-\frac{x^2}{A^2}+\frac{y^2}{B^2}=1$ be two hyperbolas having lengths of latus rectums $15 \sqrt{2}$ and $12 \sqrt{5}$ respectively. Let their eccentricities be $e_1=\sqrt{\frac{5}{2}}$ and $e_2$ respectively. If the product of the lengths of their transverse axes is $100 \sqrt{10}$,then $25 e_2^2$ is equal to . . . . . . .

The line $y = mx + c$ touches the curve $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$,if

The combined equation of the asymptotes of the hyperbola $2x^2 + 5xy + 2y^2 + 4x + 5y = 0$ is:

Let $S = \{(x,y) \in \mathbb{R}^2 : \frac{y^2}{1+r} - \frac{x^2}{1-r} = 1\}$,where $r \neq \pm 1$. Then $S$ represents

If a hyperbola has asymptotes $3x - 4y - 1 = 0$ and $4x - 3y - 6 = 0$,then the transverse and conjugate axes of that hyperbola are