Find the area of a circle that passes through the point $(4, 6)$ and has its center at $(1, 2)$. (in $\pi$)

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 $cm$)

$A$ point $P$ is taken outside $\Delta ABC$ where $B(1, \sqrt{3})$,$A(0, 0)$,and $C(2, 0)$,but inside the acute angle $BAC$,such that $\angle APC = \frac{\pi}{6}$ and $\angle BPA = \frac{\pi}{12}$. The slope of the line $BP$ 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:

$A$ line $l$ meets the circle $x^2+y^2=61$ at points $A$ and $B$. Given that $P(-5, 6)$ is a point such that $PA=PB=10$,find the equation of line $l$.

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