Question

Find the equation of a straight lines which passes through the point $(2,1)$ and makes an angle of $\pi /4$ with the straight line $2x+3y+4=0$.

Answer 1

The slope of the line $2x+3y+4=0$ is $\frac{-2}{3}$

As the line passing through $(2,3)$ and inclined to:

at $\frac{\pi }{4}\left({45}^o\right)$ i.e., ${\tan }^{-1}\left(1\right)$,

Now the slope of required line

$\tan \left(\theta \right)=\frac{\tan A-\tan B}{1+\tan A\tan B}$

$=\frac{1-\frac{2}{3}}{1+\frac{2}{3}}$

$\Rightarrow \frac{1}{5}$

$\therefore$ Equation of line passing through $(2,1)$ is

$\frac{y-1}{x-2}=\frac{1}{5}$

$5(y-1)=x-2$

$5y-5=x-2$

$5y-x-3=0$.