Question

Find the equation of those tangents to the circle $x^2+y^2-6x+4y-12=0$ which are parallel to the line $4x+3y+5=0$.

Answer 1

Let the point of contact of tangents be $(h,k)$

the tangent is $hx+ky-3(x+h)+2(y+k)-12=0⟹(h-3)x+(k+2)y+(-3h+2k-12)=0$

$h-3=\pm 4⟹h=7$ or $-1,k+2=\pm 3⟹k=1$ or $-5$

the tangent equation is $4x+3y-31=0$ or $4x+3y+19=0$