The circle$(s)$ touching the $x$-axis at a distance of $3$ from the origin and having an intercept of length $2 \sqrt{7}$ on the $y$-axis is (are):

The angle of intersection of the circles $x^2 + y^2 + 8x - 2y - 9 = 0$ and $x^2 + y^2 - 2x + 8y - 7 = 0$ is ............ $^o$.

Two concentric circles are given,where the equation of the smaller circle is $x^2 + y^2 = 4$. If each circle makes an intercept on the line $x + y = 2$ and the length of the intercept formed between the two circles is $1$,then the equation of the larger circle 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 $cm$)

If the circle $(x - h)^2 + (y - k)^2 = r^2$ touches the curve $y = x^2 + 1$ at the point $(1, 2)$,then the possible locations of the points $(h, k)$ are given by

On the circle with center $O$,points $A$ and $B$ are such that $OA = AB$. $A$ point $C$ is located on the tangent at $B$ to the circle such that $A$ and $C$ are on the opposite sides of the line $OB$ and $AB = BC$. The line segment $AC$ intersects the circle again at $F$. Then,the ratio $\angle BOF : \angle BOC$ is equal to