Question
If $$P\left( {1 + \frac{t}{{\sqrt 2 }},\,2 + \frac{t}{{\sqrt 2 }}} \right)$$ be any point on a line then the range of values of $$t$$ for which the point $$P$$ lies between the parallel lines $$x+2y=1$$ and $$2x+4y=15$$ is :
A.
$$ - \frac{{4\sqrt 2 }}{5} < t < \frac{{5\sqrt 2 }}{6}$$
B.
$$0 < t < \frac{{5\sqrt 2 }}{6}$$
C.
$$ - \frac{{4\sqrt 2 }}{5} < t < 0$$
D.
none of these
Answer :
$$ - \frac{{4\sqrt 2 }}{5} < t < \frac{{5\sqrt 2 }}{6}$$
Solution :
$$\eqalign{ & {\text{Let }}P{\text{ be on }}x + 2y = 1 \cr & \Rightarrow 1 + \frac{t}{{\sqrt 2 }} + 2\left( {2 + \frac{t}{{\sqrt 2 }}} \right) = 1{\text{ or }}t = - \frac{{4\sqrt 2 }}{3} \cr & {\text{Let }}P{\text{ be on 2}}x + 4y = 15 \cr & \Rightarrow 2\left( {1 + \frac{t}{{\sqrt 2 }}} \right) + 4\left( {2 + \frac{t}{{\sqrt 2 }}} \right) = 15{\text{ or }}t = \frac{{5\sqrt 2 }}{6} \cr} $$
$$\eqalign{ & {\text{Let }}P{\text{ be on }}x + 2y = 1 \cr & \Rightarrow 1 + \frac{t}{{\sqrt 2 }} + 2\left( {2 + \frac{t}{{\sqrt 2 }}} \right) = 1{\text{ or }}t = - \frac{{4\sqrt 2 }}{3} \cr & {\text{Let }}P{\text{ be on 2}}x + 4y = 15 \cr & \Rightarrow 2\left( {1 + \frac{t}{{\sqrt 2 }}} \right) + 4\left( {2 + \frac{t}{{\sqrt 2 }}} \right) = 15{\text{ or }}t = \frac{{5\sqrt 2 }}{6} \cr} $$