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Q.

A pair of straight lines is passing through the point (1,1). One of the lines makes an angle $θ$ with the positive direction of X-axis and the other makes the same angle with the positive direction of Y-axis. If the equation of the pair of straight lines is x² -(a + 2)xy + y² + a(x + y - 1) = 0 a $\neq$ -2 then the value of $θ$ is

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a

$\frac{1}{2} \tan (\frac{2}{a + 2})$

b

$\frac{1}{2} {Tan}^{- 1} (\frac{2}{a + 2})$

c

$\frac{1}{2} \sin (\frac{2}{a + 2})$

d

$\frac{1}{2} {Sin}^{- 1} (\frac{2}{a + 2})$

answer is A.

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Detailed Solution

$\begin{array}{l} \tan θ + \cot θ = a + 2 \Rightarrow \frac{\sin^{2} θ + \cos^{2} θ}{\cos θsin θ} = a + 2 \\ \Rightarrow \cos θsin θ = \frac{1}{a + 2} \Rightarrow \sin 2 θ = \frac{2}{a + 2} \Rightarrow θ = \frac{1}{2} {Sin}^{- 1} \frac{2}{a + 2} \end{array}$

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