First slide

Pair of straight lines

Question

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

Moderate

A

$- \frac{4 \sqrt{2}}{3} < α < \frac{5 \sqrt{2}}{6}$
B

$0 < α < \frac{5 \sqrt{2}}{6}$
C

$- \frac{4 \sqrt{2}}{3} < α < 0$
D

R

Solution

$∵ P (1 + \frac{α}{\sqrt{2}}, 2 + \frac{α}{\sqrt{2}}) lies between the parallel lines x + 2 y = 1 and 2 x + 4 y = 15, then$
$\frac{(1 + \frac{α}{\sqrt{2}}) + 2 (2 + \frac{α}{\sqrt{2}}) - 1}{2 (1 + \frac{α}{\sqrt{2}}) + 4 (2 + \frac{α}{\sqrt{2}}) - 15} < 0$
$\Rightarrow \frac{4 + \frac{3 α}{\sqrt{2}}}{- 5 + \frac{6 α}{\sqrt{2}}} < 0 \Rightarrow \frac{(α + \frac{4 \sqrt{2}}{3})}{(α - \frac{5 \sqrt{2}}{6})} < 0$
$∴ - \frac{4 \sqrt{2}}{3} < α < \frac{5 \sqrt{2}}{6}$

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