Question

If $P (1+ t / \sqrt{2}, 2+ t / \sqrt{2})$ is any point on a line, then the range of the values of t for which the point $P$ lies between the parallel lines $x+2 y=1$ and $2 x+4 y=15$ is

• A. $-4 \sqrt{2} / 3< t <5 \sqrt{2} / 6$
• B. $0< t <5 \sqrt{2} / 6$
• C. $4 \sqrt{2}< t <0$
• D. None of these

Answer 1

Answer: (A)

Let $P$ be on $x +2 y =1$. Then,

$1+\frac{t}{\sqrt{2}}+2\left(2+\frac{t}{\sqrt{2}}\right)=1 $
or $t=\frac{-4 \sqrt{2}}{3}$
Let $P$ be on $2 x +4 y =15$. Then,
$2\left(1+\frac{t}{\sqrt{2}}\right)+4\left(2+\frac{t}{\sqrt{2}}\right)=15$
or $t=\frac{5 \sqrt{2}}{6}$
Since the point lies between the lines and $x=t$.
$t \in\left(\frac{-4 \sqrt{2}}{3}, \frac{5 \sqrt{2}}{3}\right)$