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 $A(2 \sec \theta, 3 \tan \theta)$ and $B(2 \sec \phi, 3 \tan \phi)$ where $\theta+\phi=\frac{\pi}{2}$ be two points on the hyperbola $\frac{x^2}{4}-\frac{y^2}{9}=1$. If $(\alpha, \beta)$ is the point of intersection of normals to the hyperbola at $A$ and $B$, then $\beta$ is equal to

If the chord of contact of tangents from a point $P(x_1, y_1)$ to the parabola $y^2 = 4ax$ touches the parabola $x^2 = 4by$,the locus of $P$ is:

The locus of the point of intersection of the lines $\sqrt{3}x - y - 4\sqrt{3}t = 0$ and $\sqrt{3}tx + ty - 4\sqrt{3} = 0$ (where $t$ is a parameter) is a hyperbola whose eccentricity is

The eccentricity of the hyperbola $16x^{2} - 3y^{2} - 32x - 12y - 44 = 0$ is

If the latus rectum of a hyperbola subtends an angle of $120^{\circ}$ at its centre,then its eccentricity is