Lines are drawn from a point $P(-1, 3)$ to a circle $x^2 + y^2 - 2x + 4y - 8 = 0$. If the line meets the circle at two points $A$ and $B$,then the minimum value of $PA + PB$ is:

The chord of the circle $x^{2}+y^{2}-4x=0$ which is bisected at $(1,0)$ is perpendicular to the line

Find the maximum distance of the point $K(10, 7)$ from the circle $x^2 + y^2 - 4x - 2y - 20 = 0$.

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 $(1,3)$ is the midpoint of a chord of the circle $x^2+y^2-4x-8y+16=0$,then the area of the triangle formed by that chord with the coordinate axes is

Find the radius of the circle in which a central angle of $60^{\circ}$ intercepts an arc of length $37.4 \, cm$ (use $\pi = \frac{22}{7}$). (in $cm$)