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 $y+2x=5$ 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 $\frac{y-3}{x-2}=\frac{1}{0}$
$\Rightarrow 3x+4y=18$ and $x=2$