The value of $m$ for which $y = mx + 6$ is a tangent to the hyperbola $\frac{x^2}{100} - \frac{y^2}{49} = 1$ is:

The locus of the midpoints of the parallel chords with gradient $m$ of the rectangular hyperbola $xy = c^2$ is

If $(\alpha, -1)$ is an interior point of the curve $4x^2 - 3y^2 = 1$,then $\alpha$ lies in:

Let $P$ be the foot of the perpendicular from the focus $S$ of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ on the line $bx-ay=0$ and let $C$ be the centre of the hyperbola. Then,the area of the rectangle whose sides are equal to $SP$ and $CP$ is

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 $3 h^2-2 k=$

If the normal to the rectangular hyperbola $x^2-y^2=1$ at the point $P$ with parameter $\theta_1 = \frac{\pi}{4}$ meets the curve again at $Q$ with parameter $\theta_2$,then find the value of $\sec^2 \theta_2 + \tan \theta_2$.