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

If the tangent to the circle ${x^2} + {y^2} = {r^2}$ at the point $(a, b)$ meets the coordinate axes at the point $A$ and $B$, and $O$ is the origin, then the area of the triangle $OAB$ is

The angle of intersection of the circles ${x^2} + {y^2} - x + y - 8 = 0$ and ${x^2} + {y^2} + 2x + 2y - 11 = 0,$ is

Length of the tangent drawn from any point on the circle ${x^2} + {y^2} + 2gx + 2fy + {c_1} = 0$ to the circle ${x^2} + {y^2} + 2gx + 2fy + c = 0$ is

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. . . . . .

If the straight line $ax + by = 2;a,b \ne 0$ touches the circle ${x^2} + {y^2} - 2x = 3$ and is normal to the circle ${x^2} + {y^2} - 4y = 6$, then the values of a and b are respectively