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 $cm$)

Consider the following statements:
$I$. The intercept made by the circle $x^2+y^2-2x-4y+1=0$ on $Y$-axis is $2\sqrt{3}$.
$II$. The intercept made by the circle $x^2+y^2-4x-2y+6=0$ on $X$-axis is $2\sqrt{2}$.
$III$. The straight line $y=2x+1$ cuts the circle $x^2+y^2=9$ at two distinct points.
Which one of the following options is correct?
$(a)$ $I$: True,$II$: True,$III$: True
$(b)$ $I$: True,$II$: True,$III$: False
$(c)$ $I$: True,$II$: False,$III$: True
$(d)$ $I$: False,$II$: False,$III$: True

If in two circles,arcs of the same length subtend angles $60^{\circ}$ and $75^{\circ}$ at the centre,find the ratio of their radii.

The radius of any circle touching the lines $3x - 4y + 5 = 0$ and $6x - 8y - 9 = 0$ is

If $P(0,0), Q(1,0)$ and $R\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$ are three given points, then the centre of the circle for which the lines $PQ, QR$ and $RP$ are the tangents is

The length of the intercept that the circle $x^2 + y^2 + 10x - 6y + 9 = 0$ makes on the $x$-axis is: