Question

Find the lengths of the tangents drawn to the circle $x^2+y^2+2bx-3b^2=12$ from the point $(a+b,a-b).$

Answer 1

For a circle $x^2+y^2+2gx+2fy+c=0$, length of tangent from an external point $(x_1,y_1)$ is given by

$L=\sqrt{{x_1}^2+{y_1}^2+2g{x}_1+2f{y}_1+c}$

For the given circle, $x^2+y^2+2bx-3b^2=12$

$L=\sqrt{(a+b{)}^2+(a-b{)}^2+2\left(b\right)(a+b)+2\left(0\right)(a-b)-3b^2-12}$

$L=\sqrt{2a^2+b^2+2ab-12}$