Find the equation of the circle with centre $(0,2)$ and radius $2$.

Find the equation of a circle with centre $(2, 2)$ and passing through the point $(4, 5)$.

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 $3x + 4y - 5 = 0$.

Let a circle passing through $(2,0)$ have its centre at the point $(h, k)$. Let $(x_c, y_c)$ be the point of intersection of the lines $3x + 5y = 1$ and $(2+c)x + 5c^2y = 1$. If $h = \lim_{c \to 1} x_c$ and $k = \lim_{c \to 1} y_c$,then the equation of the circle is:

The equation of the circle whose diameter is the diagonal of the rectangle with sides $x=4, x=-2, y=5, y=-2$ is:

The equation of circle $C$ is: