Let $ABCD$ be a trapezium with parallel sides $AB$ and $CD$ such that the circle $S$ with $AB$ as its diameter touches $CD$. Further,the circle $S$ passes through the mid-points of the diagonals $AC$ and $BD$ of the trapezium. The smallest angle of the trapezium is

What are the coordinates of points $E$ and $F$?

If the line $x - y + 1 = 0$ is a chord of the circle $x^2 + y^2 + 4y - 4 = 0$,find the length of this chord.

Let the circles $C_1: (x-\alpha)^2 + (y-\beta)^2 = r_1^2$ and $C_2: (x-8)^2 + (y-\frac{15}{2})^2 = r_2^2$ touch each other externally at the point $(6,6)$. If the point $(6,6)$ divides the line segment joining the centres of the circles $C_1$ and $C_2$ internally in the ratio $2:1$,then $(\alpha+\beta) + 4(r_1^2 + r_2^2)$ equals

$A$ line meets the circle $x^2+y^2-4x-4y-8=0$ at two points $A$ and $B$. If $P(2,-2)$ is a point on the circle such that $PA=PB=2$,then the equation of the line $AB$ is:

If $Q$ is the inverse point of $P(-1, 1)$ with respect to the circle $x^2+y^2-2x+2y=0$,then the line containing $Q$ is