Q. Let $\theta_{1}$ and $\theta_{2}$ are the inclinations of lines $L_{1}$ and $L_{2}$ with $x$-axis. If $L_{1}\$and $L_{2}$ pass through $P\left( x_{1},\ y_{1} \right)$, then equation of one of the angle bisector of these lines is
A) $\frac{x - x_{1}}{\cos\left( \frac{\theta_{1} - \theta_{2}}{2} \right)} = \frac{y - y_{1}}{\sin\left( \frac{\theta_{1} - \theta_{2}}{2} \right)}$
B) $\frac{x - x_{1}}{{- sin}\left( \frac{\theta_{1} - \theta_{2}}{2} \right)} = \frac{y - y_{1}}{\cos\left( \frac{\theta_{1} - \theta_{2}}{2} \right)}$
C) $\frac{x - x_{1}}{\sin\left( \frac{\theta_{1} + \theta_{2}}{2} \right)} = \frac{y - y_{1}}{\cos\left( \frac{\theta_{1} + \theta_{2}}{2} \right)}$
D) $\frac{x - x_{1}}{{- sin}\left( \frac{\theta_{1} + \theta_{2}}{2} \right)} = \frac{y - y_{1}}{\sin\left( \frac{\theta_{1} + \theta_{2}}{2} \right)}$
Explanation
(d)
$y - y_{1} = \tan{\left( \frac{\theta_{1} + \theta_{2}}{2} \right)\left( x - x_{1} \right)}$
$and\ y - y_{1} = - \cot\left( \frac{\theta_{1} + \theta_{2}}{2} \right)\left( x - x_{1} \right)$
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Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.