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

Find the length of the intercept made by the circle on the $y$-axis,if the line segment joining the points $(-4, 3)$ and $(12, -1)$ is the diameter of the circle.

The equation of the circle,concentric with the circle $2x^2+2y^2-6x+8y+1=0$ and having double its area,is

$A$ circle touches the axes at the points $(3, 0)$ and $(0, -3)$. The centre of the circle is

The equation of the circle whose diameter is the line segment joining the points $(-4, 3)$ and $(12, -1)$ is:

Find the equation of the circle passing through the origin and whose center is the intersection point of the lines $2x - 3y + 4 = 0$ and $3x + 4y - 5 = 0$.