First slide

Pair of straight lines

Question

The acute angle between the pair of lines $(cosα + sinα) x^{2} - 2 xycosα + (cosα - sinα) y^{2} = 0$ is

Moderate

A

$α$
B

$2 α$
C

$\frac{α}{2}$
D

$3 α$

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 $(cosα + sinα) x^{2} - 2 xycosα + (cosα - sinα) y^{2} = 0$ , the values are $a = cosα + sinα, 2 h = - 2 cosα, b = cosα - sinα$
Hence,
$\begin{matrix} cosθ = \frac{2 cosα}{\sqrt{4 \sin^{2} α + 4 \cos^{2} α}} \\ = \frac{2 cosα}{2} \\ = cosα \end{matrix}$
Therefore, $θ = α$

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