Straight Lines
Two Lines from Point Meeting Another — θ₁+θ₂
nta_pyq_2026_jan
Grade 11
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 $\sqrt{\dfrac{2}{3}}$ from the point $P$ be $\theta_1$ and $\theta_2$. Then the value of $(\theta_1+\theta_2)$ is:
$\dfrac{\pi}{3}$
$\dfrac{\pi}{6}$
$\dfrac{\pi}{2}$
$\dfrac{\pi}{12}$
Step-by-Step Solution
Key Concept: Parametric form from $P(2,3)$: point at distance $r=\sqrt{2/3}$ is $(2+r\cos\theta,3+r\sin\theta)$. Substituting in $x+y=6$: $r(\cos\theta+\sin\theta)=1\Rightarrow\cos\theta+\sin\theta=\sqrt{3/2}$.
$\theta_1+\theta_2=\pi/2$.
Correct Answer: 3