The tangents are drawn from the point $(4, 5)$ to the circle $x^2 + y^2 - 4x - 2y - 11 = 0$. The area of the quadrilateral formed by these tangents and the radii is .............. $sq. \text{ units}$.

If one end of the diameter of the circle $x^2+y^2-4x-6y+11=0$ is $(3,4)$,then the other end of the diameter is

The intercept on the line $y=x$ by the circle $x^2+y^2-2x=0$ is $AB$. The equation of the circle with $AB$ as diameter is:

Let $n \geq 3$ and let $C_1, C_2, \ldots, C_n$ be circles with radii $r_1, r_2, \ldots, r_n$,respectively. Assume that $C_i$ and $C_{i+1}$ touch externally for $1 \leq i \leq n-1$. It is also given that the $X$-axis and the line $y=2 \sqrt{2} x+10$ are tangential to each of the circles. Then,$r_1, r_2, \ldots, r_n$ are in

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

Find the equation of the chord of the circle $x^2 + y^2 - 6x + 8y = 0$ which is bisected at the point $(5, -3)$.