Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade 11
Question:
The equation of the bisectors of the angles between the two intersecting lines $\frac{x-3}{\cos\theta} = \frac{y+5}{\sin\theta}$ and $\frac{x-3}{\cos\phi} = \frac{y+5}{\sin\phi}$ are $\frac{x-3}{\cos\alpha} = \frac{y+5}{\sin\alpha}$ and $\frac{x-3}{\cos\alpha} = \frac{y+5}{\sin\alpha}$, then:
$\alpha = \frac{\theta + \phi}{2}$
$\beta = -\sin\alpha$
$\gamma = \cos\alpha$
$\beta = \sin\alpha$
Step-by-Step Solution
Key Concept: The angle bisector of two lines through a point has slope determined by the average of the two slopes using the half-angle formula.
Two lines pass through $(3,-5)$ making angles $\theta$ and $\phi$ with the x-axis. The angle bisector $B_1$ makes angle $\alpha = \frac{\theta + \phi}{2}$ with the x-axis, so its equation is $\frac{x-3}{\cos\left(\frac{\theta+\phi}{2}\right)} = \frac{y+5}{\sin\left(\frac{\theta+\phi}{2}\right)}$. This simplifies to the angle bisector formula where the slope is determined by $\tan\left(\frac{\theta+\phi}{2}\right)$.
Correct Answer: 1,2,3