Question

Find the equations to the tangents to the circle
$x^2+y^2+2gx+2fy+c=0$ which are parallel to the line $x+2y-6=0$.

Answer 1

Let the equation of tangent be

$y=mx+d$, where $m$ is equal to the slope of line $x+2y-6=0$

$\therefore m=\frac{-1}{2}$

And we know that if a line is tangent to a circle then the distance of the centre from the line is equal to the radius of tthe circle

So, we have

$\left|\frac{-2f-g-2d}{\sqrt{5}}\right|=\sqrt{g^2+f^2-c}$

$d=\frac{\pm \sqrt{5}\sqrt{g^2+f^2-c}-g-2f}{2}$

Substituting $d$ in the equation of tangent, we get

$x+2y+g+2f=\pm \sqrt{5}\sqrt{g^2+f^2-c}$