Coordinate Geometry
Section formula; locus of dividing point on circle segment
MMTS_Full_Test_11
Grade 12
Question:
Let $P$ be a point on the line segment joining $A(5\cos\alpha,5\sin\alpha)$ and $B(5\cos\beta,5\sin\beta)$ such that $3PA=2PB$. If $|\alpha-\beta|=\dfrac{\pi}{3}$, the locus of $P$ is
(A) $x^2+y^2=13$
(B) $x^2+y^2=19$
(C) $x^2+y^2=1$
(D) $x^2+y^2=25$
Step-by-Step Solution
Key Concept: $P$ divides $AB$ in ratio $2:3$. $P=\frac{3A+2B}{5}$. $|P|^2=\frac{9|A|^2+12A\cdot B+4|B|^2}{25}=\frac{9\cdot25+12\cdot25\cos(\alpha-\beta)+4\cdot25}{25}=13+12\cos(\pi/3)=13+6=19$.
Locus: $x^2+y^2=19$.
Correct Answer: (B) $x^2+y^2=19$