Question

Find the equations to the straight lines passing through the foot of the perpendicular from the point $(h,k)$ upon the straight line $Ax+By+C=0$ and bisecting the angles between the perpendicular and the given straight line.

Answer 1

$Ax+By+C=\mathrm{0.......}\left(i\right)$

Slope of line $=m=-\frac{A}{B}$

Let the slope of perpendicular be $m^{\prime }$

$m{m}^{\prime }=-1-\frac{A}{B}{m}^{\prime }=-1\Rightarrow m^{\prime }=\frac{B}{A}$

Equation of perpendicular is

$y-k=\frac{B}{A}(x-h)Ay-Ak=Bx-BhBx-Ay+Ak-Bh=\mathrm{0......}\left(ii\right)$

Angle bisector of $\left(i\right)$ and $\left(ii\right)$ is

$\frac{Bx-Ay+Ak-Bh}{\sqrt{B^2+A^2}}=\pm \frac{Ax+By+C}{\sqrt{A^2+B^2}}Bx-Ay+Ak-Bh=\pm (Ax+By+C)B(x-h)-A(y-k)=\pm (Ax+By+C)$