Question

A line $PQ$ makes intercepts of length $2$ units between the lines $y+2x=3$ and $y+2x=5$. If the coordinates of $P$ are $(2,3),$ coordinates of $Q$ can be

• A. $(6,0)$
• B. $(2,3)$
• C. $(0,\frac{9}{2})$
• D. $(3,2)$

Answer 1

Answer: (A), (C)

The correct options are
A $(6,0)$
C $(0,\frac{9}{2})$

Since $PQ$ makes an intercept of $2$ unit between the parallel lines.
Distance between the parallel lines is (say $d$) $=\frac{5-3}{\sqrt{4+1}}$
$d=\frac{2}{\sqrt{5}}$
$\sin \theta =\frac{1}{\sqrt{5}}$ and $\tan \theta =\frac{1}{2}$
$\theta$ is the angle between the lines and the line passing through the $PQ$.
Now,

$-2$ is the slope of given parallel lines

Let the slope of line $PQ$ be $m$
$\frac{1}{2}=\frac{|-2-m|}{|1-2m|}$

$\Rightarrow \frac{-2-m}{1-2m}=\pm \frac{1}{2}$

$\Rightarrow \frac{-2-m}{1-2m}=\frac{1}{2}and\frac{-2-m}{1-2m}=-\frac{1}{2}$

$\Rightarrow \frac{-2-m}{1-2m}=\frac{1}{2}$, if you solve this you don't get the value of $m$

So, on solving $\frac{-2-m}{1-2m}=-\frac{1}{2}$

we get $m=\frac{-3}{4}$

$\frac{-3}{4}=\frac{y-3}{x-2}$

Verifying through options, we get $Q$ is $(6,0)$ and $(0,\frac{9}{2})$

Hence, option A and C.