The locus of the midpoints of the chords of the circle $x^2 + y^2 = a^2$ which touch the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ 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 $3 h^2-2 k=$

The equation of a line passing through the centre of a rectangular hyperbola is $x - y - 1 = 0$. If one of the asymptotes is $3x - 4y - 6 = 0$,the equation of the other asymptote is:

The length of the conjugate axis of a hyperbola is greater than the length of the transverse axis. Then,the eccentricity $e$ is

Assertion: The distance between the points $P(\frac{\pi}{4})$ and $P(\frac{\pi}{3})$ on the hyperbola $9x^2 - 16y^2 = 9$ is $\frac{1}{4} \sqrt{66 - 32\sqrt{2} - 18\sqrt{3}}$.
Reason: $x = a \cosh t, y = b \sinh t$ are the parametric equations of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$.

The circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ intersect at points $A$ and $B$. The line $2x + y = 1$ is a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. If this line passes through the point of intersection of the nearest directrix and the $x$-axis,find the eccentricity of the hyperbola.