First slide

Pair of straight lines

Question

If $θ$ is acute angle between two lines $x^{2} + 6 xy + y^{2} = 0$ then the value of $\sqrt{2} sinθ + cosθ$ is

Easy

A

$- \frac{3}{5}$
B

$\frac{3}{5}$
C

$- \frac{5}{3}$
D

$\frac{5}{3}$

Solution

If $θ$ is acute angle between pair of lines represented by the equation ${ax}^{2} + 2 hxy + {by}^{2} = 0$ then $cosθ = \frac{|a + b|}{\sqrt{{(a - b)}^{2} + {(2 h)}^{2}}}$
For the equation $x^{2} + 6 xy + y^{2} = 0, a = 1, h = 3, b = 1$
Hence,
$\begin{matrix} cosθ = \frac{2}{6} \\ = \frac{1}{3} \end{matrix}$

Use the right triangle for $sinθ = \frac{2 \sqrt{2}}{3}$
Hence,
$\begin{matrix} \sqrt{2} \sin θ + \cos θ = \sqrt{2} (\frac{2 \sqrt{2}}{3}) + \frac{1}{3} \\ = \frac{4}{3} + \frac{1}{3} \\ = \frac{5}{3} \end{matrix}$
Therefore, $\sqrt{2} \sin θ + \cos θ = \frac{5}{3}$

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