Parabola
Locus of Mid-Point Chord Then Internal Division
nta_pyq_2026_jan
Grade 11
Question:
Let the locus of the mid-point of the chord through the origin $O$ of the parabola $y^2=4x$ be the curve $S$. Let $P$ be any point on $S$. Then the locus of the point, which internally divides $OP$ in the ratio $3:1$, is:
2x^2=3y
3y^2=2x
2y^2=3x
3x^2=2y
Step-by-Step Solution
Key Concept: For chord through $O$ with end $(t^2,2t)$: mid-point $M=(t^2/2,t)$. Locus $S$: $k^2=2h\Rightarrow y^2=2x$. Point on $S$: $Q=(t^2/2,t)$. $R$ divides $OQ$ in $3:1$: $R=(3t^2/8, 3t/4)$.
Locus: $2y^2=3x$.
Correct Answer: 3