Question

Find the equations of those tangents to the circle $x^2+y^2=52$ which are parallel to the straight line $2x+3y=6$.

Answer 1

Equation of the tangent to the circle is given by

$y=mx\pm a\sqrt{1+m^2}$

In this case, $a=\sqrt{52}=2\sqrt{13}$ and $m=\frac{-2}{3}$

$\therefore$ equations of tangent are

$y=\frac{-2}{3}x\pm 2\sqrt{13}\times \sqrt{1+\frac{4}{9}}$

$\Rightarrow 3y+2x=\pm 2\left(13\right)=\pm 26$

$\therefore 3y+2x=26$ and $3y+2x=-26$ are the equations of tangents.