Find the equation of the circle concentric with the circle $x^2 + y^2 - 4x - 6y - 3 = 0$ and which touches the Y axis.
Equation of given circle: $\hspace{1em}$ $x^2 + y^2 - 4x - 6y - 3 = 0$
Comparing with the standard equation of circle: $\hspace{1em}$ $x^2 + y^2 + 2gx + 2fy + c = 0$ $\;$ gives
$2g = -4 \implies g = -2$, $\;\;$ $2f = -6 \implies f = -3$, $\;\;$ $c = -3$
$\therefore$ $\;$ Center of given circle $= C = \left(-g,-f\right) = \left(2,3\right)$
$\because$ $\;$ The required circle is concentric with the given circle,
$\therefore$ $\;$ center of required circle $= C = \left(2,3\right)$
Let the required circle touch the Y axis at the point $A$.
Then, $A = \left(0,3\right)$
Radius of the required circle $= CA = \sqrt{\left(2 - 0\right)^2} = 2$
$\therefore$ $\;$ Equation of required circle is
$\left(x - 2\right)^2 + \left(y - 3\right)^2 = 2^2$
i.e. $x^2 + y^2 - 4x - 6y + 4 + 9 = 4$
i.e. $x^2 + y^2 - 4x - 6y + 9 = 0$