Straight Lines
Locus of Division Point on Chord of Circle
nta_pyq_2023_apr
Grade 11
Question:
A chord $AB$ of length $\lambda$ moves so that $A,B$ are on a circle of radius $\lambda$. Locus of point dividing $AB$ in ratio $2:3$ is a circle of radius
\dfrac{3\lambda}{5}
\dfrac{2\lambda}{3}
\dfrac{\sqrt{19}\lambda}{5}
\dfrac{\sqrt{19}\lambda}{7}
Step-by-Step Solution
Key Concept: $A=(\lambda\cos\alpha,\lambda\sin\alpha)$, $B=(\lambda\cos\beta,\lambda\sin\beta)$. $P$ divides $AB$ in $2:3$: $P=(\frac{3A+2B}{5})$. $|OP|^2=\frac{1}{25}(9|A|^2+4|B|^2+12A\cdot B)=\frac{13\lambda^2}{25}+\frac{12\lambda^2\cos(\alpha-\beta)}{25}$.
Radius $=\frac{\sqrt{19}\lambda}{5}$.
Correct Answer: 3