The area of the quadrilateral formed by the tangents from the point $(4,5)$ to the circle $x^2+y^2-4x-2y-11=0$,and the pair of radii joining the points of contact of these tangents is

If the area of an equilateral triangle inscribed in the circle $x^2 + y^2 + 10x + 12y + c = 0$ is $27\sqrt{3} \text{ sq. units}$,then $c$ is equal to

The area of a circle whose centre is $(h, k)$ and radius $a$ is

$A$ rectangle $ABCD$ is inscribed in a circle whose center lies on the line $3y = x + 10$. If $A$ and $B$ are the points $(-6, 7)$ and $(4, 7)$ respectively,find the area of the rectangle.

If $(a, b)$ is the centre of the circle passing through the vertices of the triangle formed by $x+y=6, 2x+y=4$ and $x+2y=5$,then $(a, b)$ is

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