The number of common tangents to the circles ${x^2} + {y^2} - x = 0$ and ${x^2} + {y^2} + x = 0$ is:

$A$ square is inscribed in the circle $x^2+y^2-2x+4y-93=0$ with its sides parallel to the coordinate axes. Which of the following can be one of the vertices of the square?

Suppose $S_1$ and $S_2$ are two unequal circles,$AB$ and $CD$ are the direct common tangents to these circles. $A$ transverse common tangent $PQ$ cuts $AB$ in $R$ and $CD$ in $S$. If $AB=10$,then $RS$ is

The number of common tangents to the circles $x^2 + y^2 = 1$ and $x^2 + y^2 - 4x + 3 = 0$ is

When is the angle of intersection of two circles $0^{\circ}$?

Consider a semicircle of radius $1$ unit constructed on the diameter $AB$ and let $O$ be its centre. Let $C$ be a point on $AO$ such that $AC:CO = 2:1$. Draw $CD$ perpendicular to $AO$ with $D$ on the semicircle. Draw $OE$ perpendicular to $AD$ with $E$ on $AD$. Let $OE$ and $CD$ intersect at $H$. Then,$DH$ equals