The number of real circles cutting orthogonally the circle $x^{2}+y^{2}+2x-2y+7=0$ is

The locus of the centre of the circle which touches the circle ${x^2} + {(y - 1)^2} = 1$ externally and also touches the $x$-axis is

$A$ straight line passing through the point $(1,0)$ and not parallel to the $x$-axis intersects the curve $2x^2+5y^2-7x=0$ at two points $A$ and $B$. The angle subtended by the line segment $AB$ at the origin is (in $^\circ$)

$A$ circle with centre $P$ is tangent to the negative $x$-axis and negative $y$-axis and is externally tangent to a circle with centre $(-6, 0)$ and radius $2$. What is the sum of all possible radii of the circle with centre $P$?

The diameter of the circle $x^2 + y^2 - 2x - 6y + 6 = 0$ is a chord of another circle with center $(2, 1)$. Find the radius of this circle.

$A(x_1, y_1)$ 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: