The coordinates of any point on the circle $x^2 + y^2 = 4$ are:

Circles are drawn through the point $(2,0)$ to cut intercepts of length $5$ units on the $X$-axis. If their centre lies in the first quadrant,then their equation is

Let a circle $C$ have its centre in the first quadrant, intersect the coordinate axes at exactly three points and cut off equal intercepts from the coordinate axes. If the length of the chord of $C$ on the line $x + y = 1$ is $\sqrt{14}$, then the square of the radius of $C$ is . . . . . .

The equation of a circle of radius $5$ units touching another circle $x^2+y^2-2x-4y-20=0$ at $(5,5)$ is

If a circle passing through $(4, -2)$ is concentric with the circle $x^2 + y^2 - 2x + 4y + 20 = 0$,then find the value of $c$ for the circle $x^2 + y^2 - 2x + 4y + c = 0$.

The equation of the circle with centre at $(1, -2)$ and passing through the centre of the given circle $x^2 + y^2 + 2y - 3 = 0$ is