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

$A$ variable circle passes through the fixed point $A(p, q)$ and touches the $x$-axis. The locus of the other end of the diameter through $A$ is

$P$ and $Q$ are the ends of a diameter of the circle $x^2+y^2=a^2$ where $a > \frac{1}{\sqrt{2}}$. $s$ and $t$ are the lengths of the perpendiculars drawn from $P$ and $Q$ onto the line $x+y=1$ respectively. When the product $st$ is maximum,the greater value among $s$ and $t$ is

$ABCD$ is a square with side length $a$. Taking $AB$ and $AD$ as the coordinate axes,find the equation of the circle passing through the vertices of the square.

If the two circles,$x^2 + y^2 + 2g_1x + 2f_1y = 0$ and $x^2 + y^2 + 2g_2x + 2f_2y = 0$ touch each other,then:

Let $y^2 = 4ax$ be a parabola and $x^2 + y^2 + 2bx = 0$ be a circle. If the parabola and the circle touch each other externally,then: