The number of common tangents to the circles $x^2+y^2+2x+8y-23=0$ and $x^2+y^2-4x-10y+19=0$ is:

If two distinct chords,drawn from the point $(p, q)$ on the circle ${x^2} + {y^2} = px + qy$ (where $pq \neq 0$),are bisected by the $x$-axis,then:

Let $OA$ be a radius of a circle with center $O$ and radius $d$. Let $B$ be a point on the circle such that $\angle AOB = \theta$ $(< \frac{\pi}{2})$. Let $D$ be a point on $OA$ such that $BD \perp OA$. Let $E$ be the mid-point of $BD$ and $F$ be a point on the arc $AB$ such that $EF \parallel OA$. Then,the ratio of the length of the arc $AF$ to the length of the arc $AB$ is

If a circle with center $(-1, 1)$ touches the line $x + 2y + 4 = 0$, then the coordinates of the point of contact are

Chords $AB$ and $CD$ of a circle intersect at a right angle at point $P$. If the lengths of $AP$, $PB$, $CP$, and $PD$ are $2$, $6$, $3$, and $4$ units respectively, then the radius of the circle is:

Let $y=x$ be the equation of a chord of the circle $C_{1}$ (in the closed half-plane $x \ge 0$) of diameter $10$ passing through the origin. Let $C_{2}$ be another circle described on the given chord as its diameter. If the equation of the chord of the circle $C_{2}$, which passes through the point $(2, 3)$ and is farthest from the center of $C_{2}$, is $x+ay+b=0$, then $a-b$ is equal to: