Circles
Locus of Midpoint of Chord — Subtended Angle
nta_pyq_2023_jan
Grade None
Question:
The locus of the mid points of the chords of the circle $C:(x-4)^2+(y-5)^2=4$ which subtend an angle $\theta_1$ at the centre of circle $C_1$, is a circle of radius $r_1$. If $\theta_1=\dfrac{\pi}{3}$, $\theta_3=\dfrac{2\pi}{3}$ and $r_1^2=r_2^2+r_3^2$, then $\theta_2$ is equal to:
$\dfrac{\pi}{4}$
$\dfrac{3\pi}{4}$
$\dfrac{\pi}{6}$
$\dfrac{\pi}{2}$
Step-by-Step Solution
Key Concept: For a chord subtending angle $\theta$ at centre of circle radius $R$: midpoint is at distance $R\cos(\theta/2)$ from centre. Locus is circle of radius $r=R\cos(\theta/2)$. Here $R=2$.
$\theta_2=\dfrac{\pi}{2}$.
Correct Answer: 4