Point from which two distinct tangents can be drawn on two different branches of the hyperbola $\frac{{{x^2}}}{{25}} – \frac{{{y^2}}}{{16}} = \,1$ but no two different tangent can be drawn to the circle $x^2 + y^2 = 36$ is

The equation of the hyperbola in the standard form (with transverse axis along the $x$ –  axis) having the length of the latus rectum = $9$ units and eccentricity = $5/4$ is

If the line $x-1=0$, is a directrix of the hyperbola $kx ^{2}- y ^{2}=6$, then the hyperbola passes through the point.

Length of latusrectum of curve $xy = 7x + 5y$ is

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 $1, 2$