Find the length of the tangent from the point $\left(1,2\right)$ to the circle $x^2 + y^2 -2x + 4y + 9 = 0$.
Comparing the given circle: $\;\;\;$ $x^2 + y^2 -2x + 4y + 9 = 0$
with the standard equation of circle: $\;\;\;$ $x^2 + y^2 + 2gx + 2fy + c = 0$
gives: $\;\;\;$ $2g = -2$ $\implies$ $g = -1$; $\;\;$ $2f = 4$ $\implies$ $f = 2$; $\;\;$ $c = 9$
Length of tangent from an external point $P \left(x_1, y_1\right)$ and touch the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ at point $T$ is
$PT = \sqrt{x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c}$
$\therefore$ $\;$ Required length of tangent from $\left(1,2\right)$ to the given circle
$= \sqrt{1^2 + 2^2 - 2 \times 1 + 4 \times 2 + 9}$
$= \sqrt{1 + 4 - 2 + 8 + 9} = \sqrt{20} = 2 \sqrt{5}$ units