When does the equation $x^2 + y^2 + 2gx + 2fy + c = 0$ represent a point circle?

Find the equation of the circle with radius $5$ whose centre lies on the $x-$axis and passes through the point $(2, 3)$.

The equation of the circle with the origin as the centre passing through the vertices of an equilateral triangle whose median is of length $3a$ is

In $\triangle ABC$,$\angle A = 90^{\circ}$ and the coordinates of points $B$ and $C$ are $(2, -4)$ and $(1, 5)$. Then the equation of the circumcircle of $\triangle ABC$ is

Find the centre and radius of the circle $2x^{2} + 2y^{2} - x = 0$.

The equation of a circle with centre $(-4, 3)$ and touching the circle $x^2 + y^2 = 1$ is