The eccentricity of curve ${x^2} – {y^2} = 1$ is

The vertices of a hyperbola $H$ are $(\pm 6,0)$ and its eccentricity is $\frac{\sqrt{5}}{2}$. Let $N$ be the normal to $H$ at a point in the first quadrant and parallel to the line $\sqrt{2} x + y =2 \sqrt{2}$. If $d$ is the length of the line segment of $N$ between $H$ and the $y$-axis then $d ^2$ is equal to $…………$.

The circle $x^2+y^2-8 x=0$ and hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$ intersect at the points $A$ and $B$

$2.$ Equation of a common tangent with positive slope to the circle as well as to the hyperbola is

$(A)$ $2 x-\sqrt{5} y-20=0$ $(B)$ $2 x-\sqrt{5} y+4=0$

$(C)$ $3 x-4 y+8=0$ $(D)$ $4 x-3 y+4=0$

$2.$ Equation of the circle with $\mathrm{AB}$ as its diameter is

$(A)$ $x^2+y^2-12 x+24=0$ $(B)$ $x^2+y^2+12 x+24=0$

$(C)$ $\mathrm{x}^2+\mathrm{y}^2+24 \mathrm{x}-12=0$ $(D)$ $x^2+y^2-24 x-12=0$

Give hte answer question $1, 2$

The curve $xy = c, (c > 0)$, and the circle $x^2 + y^2 = 1$ touch at two points. Then the distance between the points of contacts is

The equation of the hyperbola whose conjugate axis is $5$ and the distance between the foci is $13$, is