Mathematics · Parabola

JEE Main 2026 — 4 April, Evening Shift — Question 48

Let A, B and C be the vertices of a variable right angled triangle inscribed in the parabola $y^2 = 16x$ . Let the vertex B containing the right angle be (4,8) and the locus of the centroid of $\triangle ABC$ be a conic $C_0$ . Then three times the length of latus rectum of $C_0$ is

Answer: 16

Numerical answer — enter this value.

Step-by-step solution

Variable point A & C are shown in the figure $\because \mathrm{m}_{\mathrm{AB}} \cdot \mathrm{m}_{\mathrm{BC}}=-1$ $\Rightarrow \mathrm{t}_{1}+\mathrm{t}_{2}+\mathrm{t}_{1} \mathrm{t}_{2}=-5 \ldots(1)$

Suppose locus of centroid of $\triangle \mathrm{ABC}$ is ( $\mathrm{h}, \mathrm{k}$ ) $\therefore 3 \mathrm{~h}=4+4 \mathrm{t}_{1}{ }^{2}+4 \mathrm{t}_{2}{ }^{2} \& 3 \mathrm{k}=8+8 \mathrm{t}_{1}+8 \mathrm{t}_{2}$ by eliminating $\mathrm{t}_{1} \& \mathrm{t}_{2}$ using equation (1) also we get $h=\frac{9}{48} k^{2}+\frac{40}{3}$ ∴ locus of centroid of parabola is $\mathrm{x}=\frac{9}{48} \mathrm{y}^{2}+\frac{40}{3}$ $\therefore \ell($ L.R. $)=\frac{48}{9}$ $\therefore 3 \ell($ L.R. $)=\frac{48}{3}=16$

Answer key and solution verified before publishing.

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Exam: JEE Main 2026
Paper: 4 April, Evening Shift
Subject: Mathematics
Chapter: Parabola
Topic: Introduction to Parabola