Question

Find the equation of the line which bisects the obtuse angle between the lines
$x-2y+4=0$ and $4x-3y+2=0$

Answer 1

$x-2y+4=0,4x-3y+2=0$
$c_1$ and $c_2$ are both +ive and hence taking + out of $\pm$ signs we shall get the bisector of the angle in which origin lies. Again
$a_1{a}_2+b_1{b}_2=4+6=10$, +ive
Therefore origin lies in obtuse angle
$\frac{x-2y+4}{\sqrt{5}}=+\frac{4x-3y+2}{5}.....\left(1\right)$
is the bisector of angle in which origin lies and it lies in obtuse angle. Its equation is
$(4-\sqrt{5})x-(3-2\sqrt{5})y+(2-4\sqrt{5})=0$
Hence the other bisector of acture angle whose equation is
$(4+\sqrt{5})x-(3+2\sqrt{5})y+(2+4\sqrt{5})=0$