Question

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

Answer 1

Given circle is $x^2+y^2-6x+4y-12=\mathrm{0.........................}\left(i\right)$
and given line is $4x+3y+5=0$ ......................(ii)
Centre of circle (i) is $(3,2)$ and its radius is $5$ Equation of any ine
$4x+3y+k=0$ parallel to the line (ii) ................(iii)
If line (iii) is tangent to circle (i) then
$\frac{|4.3+3(-2)+k|}{\sqrt{4^2+3^2}}=5or|6+k|=25$
or $6+k=$ $\pm 25\therefore k=19,-31$ Hence equation of required tangents are $4x+3y+19=0$ and $4x+3y-31=0$