If a circle $C$ passing through the point $(4, 0)$ touches the circle $x^2 + y^2 + 4x -6y = 12$ externally at the point $(1, -1)$, then the radius of $C$ is

In the figure, $A B C D$ is a unit square. A circle is drawn with centre $O$ on the extended line $C D$ and passing through $A$. If the diagonal $A C$ is tangent to the circle, then the area of the shaded region is

If a circle of radius $R$ passes through the origin $O$ and intersects the coordinate axes at $A$ and $B,$ then the locus of the foot of perpendicular from $O$ on $AB$ is

A tangent $P T$ is drawn to the circle $x^2+y^2=4$ at the point $P(\sqrt{3}, 1)$. A straight line $L$, perpendicular to $P T$ is a tangent to the circle $(x-3)^2+y^2=1$.

$1.$ A common tangent of the two circles is

$(A)$ $x=4$ $(B)$ $y=2$ $(C)$ $x+\sqrt{3} y=4$ $(D)$ $x+2 \sqrt{2} y=6$

$2.$ A possible equation of $L$ is

$(A)$ $x-\sqrt{3} y=1$ $(B)$ $x+\sqrt{3} y=1$ $(C)$ $x-\sqrt{3} y=-1$ $(D)$ $x+\sqrt{3} y=5$

Give the answer question $1$ and $2.$

The equation of the normal at the point $(4,-1)$ of the circle $x^2+y^2-40 x+10 y=153$ is

The equation of three circles are ${x^2} + {y^2} - 12x - 16y + 64 = 0,$ $3{x^2} + 3{y^2} - 36x + 81 = 0$ and ${x^2} + {y^2} - 16x + 81 = 0.$ The co-ordinates of the point from which the length of tangent drawn to each of the three circle is equal is