Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade None
Question:
Two points $P_1$ and $P_2$ are at distances $r_1$ and $r_2$ respectively from the origin $O$ and $OP_1$ and $OP_2$ makes angle $\theta_1$ and $\theta_2$ respectively with the x-axis. Let there be a point $P$ on $P_1P_2$ such that $OP$ makes an angle $\frac{\theta_2 + \theta_1}{2}$ with the x-axis. Then $OP$ is:
$\frac{2r_1r_2}{r_1 + r_2}\cos\frac{\theta_2 - \theta_1}{2}$
$\frac{2r_1r_2}{r_1 + r_2}\sin\frac{\theta_2 - \theta_1}{2}$
$\frac{r_1r_2}{r_1 + r_2}\cos\frac{\theta_2 + \theta_1}{2}$
$\frac{r_1r_2}{r_1 + r_2}\sin\frac{\theta_2 + \theta_1}{2}$
Step-by-Step Solution
Key Concept: Areas of sub-triangles formed by intercepts are additive when sharing a common vertex.
The sum of areas of triangles $\triangle OQP_1$ and $\triangle OQP_2$ equals the area of $\triangle OQP_S$. This is a property of how a line divides regions: when a line intersects two axes at points creating two smaller triangles with a common vertex, their combined area relates to the total triangular region formed.
Correct Answer: 1