The tangent and the normal lines at the point $(\sqrt 3,1)$ to the circle $x^2 + y^2 = 4$ and the $x -$ axis form a triangle. The area of this triangle (in square units) is

The equation of the tangent to the circle ${x^2} + {y^2} = {a^2}$ which makes a triangle of area ${a^2}$ with the co-ordinate axes, is

If the length of tangent drawn from the point $(5, 3)$ to the circle ${x^2} + {y^2} + 2x + ky + 17 = 0$ be $7$, then $k$ =

Let the point $B$ be the reflection of the point $A(2,3)$ with respect to the line $8 x-6 y-23=0$. Let $\Gamma_A$ and $\Gamma_B$ be circles of radii $2$ and $1$ with centres $A$ and $B$ respectively. Let $T$ be a common tangent to the circles $\Gamma_A$ and $\Gamma_B$ such that both the circles are on the same side of $T$. If $C$ is the point of intersection of $T$ and the line passing through $A$ and $B$, then the length of the line segment $AC$ is. . . . . .

The value of $c$, for which the line $y = 2x + c$ is a tangent to the circle ${x^2} + {y^2} = 16$, is

Let the tangents at two points $A$ and $B$ on the circle $x ^{2}+ y ^{2}-4 x +3=0$ meet at origin $O (0,0)$. Then the area of the triangle of $OAB$ is.