Let $A_0 A_1 A_2 A_3 A_4 A_5$ be a regular hexagon inscribed in a circle of unit radius. The product of the lengths of the line segments $A_0A_1$,$A_1A_2$,and $A_0A_4$ is

If the line $x - y + 1 = 0$ is a chord of the circle $x^2 + y^2 + 4y - 4 = 0$,find the length of this chord.

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

Two circles $(x + a)^2 + (y + b)^2 = a^2$ and $(x + \alpha)^2 + (y + \beta)^2 = \beta^2$ cut orthogonally if:

Three circles of radius $r$ touch each other externally. What is the radius of the circle that touches all three circles internally?

The square of the length of the tangent from the point $(3, -4)$ to the circle $x^2 + y^2 - 4x - 6y + 3 = 0$ is ....