Two circles $(x + a)^2 + (y + b)^2 = a^2$ and $(x + \alpha)^2 + (y + \beta)^2 = \beta^2$ cut orthogonally if:

If $A(1,2)$ and $B(2,1)$ are two vertices of an acute-angled triangle and $S(0,0)$ is its circumcentre,then the angle subtended by $AB$ at the third vertex is

$AB$ is a line segment of length $24 \ cm$ and $C$ is its midpoint. On $AB$,two semicircles with diameters $AC$ and $CB$ are described on the same side. $A$ large semicircle with diameter $AB$ is also described on the same side. Find the radius of the circle that touches all three semicircles. (in $cm$)

The line $y=mx+c$ intersects the circle $x^2+y^2=r^2$ at two distinct points if:

The length of the chord intercepted by the circle $x^2+y^2-4x+4y+3=0$ on the line $x=3y+13$ is units.

$A$ rectangle $ABCD$ is inscribed in a circle whose center lies on the line $3y = x + 10$. If $A$ and $B$ are the points $(-6, 7)$ and $(4, 7)$ respectively,find the area of the rectangle.