Question

Find the equations of the straight lines passing through the point $(2,3)$ and inclined at $\frac{\pi }{4}$ radians to the line $2x+3y=5$.

Answer 1

Let the line $2x+3y=5$ makes angle $\theta$ with positive $x-$axis. Then $\tan \theta =\frac{-2}{3}$

Slope of the required lines are

$\tan (\theta +\frac{\pi }{4})=\frac{\tan \theta +\tan \frac{\pi }{4}}{1-\tan \theta \tan \frac{\pi }{4}}=\frac{\frac{-2}{3}+1}{1-\left(\frac{-2}{3}\right)}=\frac{\frac{1}{3}}{\frac{3+2}{3}}=\frac{1}{5}$

and $\tan (\theta -\frac{\pi }{4})=-5$

$\therefore$ the equations of the required lines are

$y-3=\frac{1}{5}(x-2)$ or $x-5y+13=0$

$y-3=-5(x-2)$ or $5x+y-13=0$