Question

Find the equations of the line passing through the point $(2, 3)$ and making intercept of length $3$ unit between the lines $y + 2x = 2$ and $y + 2x = 5$.

Answer 1

Let $I$ makes an angle a with the given parallel lines and intercept $AB$ is of $3$ units.

Now, distance between parallel lines
$=\frac{|5-2|}{\sqrt{1^{2}+2^{2}}}=\frac{3}{\sqrt{5}}$
$\therefore \sin \alpha=\frac{1}{\sqrt{5}}, \cos \alpha=\frac{2}{\sqrt{5}}$
and $ \tan \alpha=\frac{1}{2}$
$\Rightarrow$ Equation of straight line passing through $(2,3)$ and
making an angle $\alpha$ with $y+2 x=5$ is
$\frac{y-3}{x-2}=\tan (\theta+\alpha) \Rightarrow \frac{y-3}{x-2}=\frac{\tan \theta+\tan \alpha}{1-\tan \theta \tan \alpha}$
and $ \frac{y-3}{x-2}=\frac{\tan \theta-\tan \alpha}{1+\tan \theta \tan \alpha}$
$\Rightarrow \frac{y-3}{x-2}=-\frac{3}{4}$ and $\frac{y-3}{x-2}=\frac{1}{0}$
$\Rightarrow 3 x+4 y=18$ and $x=2$