Find the equation of the pair of tangents drawn from the origin to the circle $x^2 + y^2 + 20(x + y) + 20 = 0$.

$A$ rhombus is inscribed in the region common to the two circles $x^2 + y^2 - 4x - 12 = 0$ and $x^2 + y^2 + 4x - 12 = 0$,with two of its vertices on the line joining the centres of the circles. The area of the rhombus is:

The equation of the circle symmetric to the circle $x^2 + y^2 - 2x - 4y + 4 = 0$ about the line $x - y = 3$ is

If the variable line $3x + 4y = \alpha$ lies between the two circles $(x - 1)^2 + (y - 1)^2 = 1$ and $(x - 9)^2 + (y - 1)^2 = 4$ without intercepting a chord on either circle,then the sum of all the integral values of $\alpha$ is .... .

If two circles of the same radius $a$ and centers at $(2, 3)$ and $(5, 6)$ are orthogonal,find the value of $a$.

$A(x_1, y_1)$ is the internal centre of similitude and $B(x_2, y_2)$ is the external centre of similitude of two circles $C_1$ and $C_2$ whose centres are $P(\alpha, \beta)$ and $Q(\gamma, \delta)$ respectively. If $PA=3, AB=5, QB=2$,then the ratio of the radii of the two circles is: