Straight LineMedium
Question
Let the angles made with the positive x -axis by two straight lines drawn from the point $P(2,3)$ and meeting the line $x + y = 6$ at a distance $\sqrt{\frac{2}{3}}$ from the point P be $\theta_{1}$ and $\theta_{2}$. Then the value of $\left( \theta_{1} + \theta_{2} \right)$ is :
Options
A.$\frac{\pi}{12}$
B.$\frac{\pi}{6}$
C.$\frac{\pi}{2}$
D.$\frac{\pi}{3}$
Solution
Let Q is $\left( \sqrt{\frac{2}{3}}cos\theta + 2,\sqrt{\frac{2}{3}}sin\theta + 3 \right)$
so, $x + y = 6$
$${\sqrt{\frac{2}{3}}(cos\theta + sin\theta) + 5 = 6 }{sin\theta + cos\theta = \sqrt{\frac{3}{2}} }{1 + sin2\theta = \frac{3}{2} }{sin2\theta = \frac{1}{2} }{2\theta = \frac{\pi}{6},\frac{5\pi}{6} }{\theta = \frac{\pi}{12}\&\frac{5\pi}{6} }$$So $\theta_{1} + \theta_{2} = \frac{\pi}{2}$
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