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 $x+2y=1$ and $2x+4y=15$ 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 $x+2y=1$. 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 $2x+4y=15$. 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 $x=t$.
$t\in \left(\frac{-4\sqrt{2}}{3},\frac{5\sqrt{2}}{3}\right)$