Question

Find the equations of the lines which pass through the point of intersection of the lines $4x-3y-1=0$ and $2x-5y+3=0$ and are equally inclined to the axes.

Answer 1

$\Rightarrow$ Line $l$ would be of the form

$l=L_1+\lambda L_2=0$

$\Rightarrow (4x-3y-1)+\lambda (2x-5y+3)=0$

$\Rightarrow (4+2\lambda )x+(-3-5\lambda )y-1+3\lambda =0$

line equally inclined to

axes has slope $\theta ={45}^{\circ }$

$\Rightarrow 2\theta ={90}^{\circ }$

$\Rightarrow \frac{-(4+2\lambda )}{(-3-5\lambda )}=1$ $\{-\frac{A}{B}\}$

$\Rightarrow 4+2\lambda =3+5\lambda$

$\Rightarrow 3\lambda =1\Rightarrow \lambda =\frac{1}{3}$

$\Rightarrow l:(4+\frac{2}{3})x+(-3-\frac{5}{3})y-1+1=0$

$\Rightarrow \frac{14}{3}x-\frac{14}{3}y=0$

$\Rightarrow x-y=0$