Straight Lines
Locus of Midpoints of Intercepts
nta_pyq_2023_apr
Grade None
Question:
If $(\alpha,7\sqrt{3})$ lies on the curve traced by midpoints of $x\cos\theta+y\sin\theta=7$, $\theta\in(0,\frac{\pi}{2})$ between axes, then $\alpha$ is equal to
$-7$
$-7\sqrt{3}$
$7\sqrt{3}$
$7$
Step-by-Step Solution
Key Concept: Intercepts: $(\frac{7}{\cos\theta},0)$ and $(0,\frac{7}{\sin\theta})$. Midpoint: $(h,k)=(\frac{7}{2\cos\theta},\frac{7}{2\sin\theta})$. Locus: $\frac{1}{h^2}+\frac{1}{k^2}=\frac{4}{49}$.
$\alpha=7$.
Correct Answer: 4