First slide

Pair of straight lines

Question

If the pair of lines represented by the combined equation $2 x^{2} + 4 xy + y^{2} = 0$ makes angles $θ_{1}, θ_{2}$ with $x -$ axis then $\cos (θ_{1} - θ_{2}) =$

Easy

A

$\frac{3}{\sqrt{17}}$
B

$\frac{\sqrt{17}}{3}$
C

$\sqrt{\frac{3}{17}}$
D

$\sqrt{\frac{17}{3}}$

Solution

If $θ_{1}, θ_{2}$ are the inclination of the lines represented by the combined equation $2 x^{2} + 4 xy + y^{2} = 0$ then $θ_{1} - θ_{2}$ is the angle between those two lines.
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 $2 x^{2} + 4 xy + y^{2} = 0, a = 2, h = 2, b = 1$ ,
Hence,
$\begin{matrix} \cos (θ_{1} - θ_{2}) = \frac{3}{\sqrt{1 + 16}} \\ = \frac{3}{\sqrt{17}} \end{matrix}$
Therefore, the required value is $\cos (θ_{1} - θ_{2}) = \frac{3}{\sqrt{17}}$

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