JEE Main 2026 — Apr 4, Shift 2 — Mathematics solution

JEE Main2026Apr 4, Shift 2Mathematics

Let $A$ , $B$ and $C$ be the vertices of a variable right angled triangle inscribed in the parabola $y^{2} = 16 x$ . Let the vertex $B$ containing the right angle be $(4, 8)$ and the locus of the centroid of $△ A B C$ be a conic $C_{0}$ . Then three times the length of latus rectum of $C_{0}$ is ______.

Your answer

Solution

Variable point $A$ and $C$ are shown in the figure

$\therefore\; m_{AB}\cdot m_{BC}=-1$

$\Rightarrow\; t_1+t_2+t_1t_2=-5 \qquad \cdots (1)$

Suppose locus of centroid of $△ A B C$ is $(h, k)$

$\therefore\; 3h=4+4t_1^2+4t_2^2 $ and $3 k = 8 + 8 t_{1} + 8 t_{2}$

By eliminating $t_{1}$ and $t_{2}$ using equation (1), also we get

$h = \frac{9}{48} k^{2} + \frac{40}{3}$

$∴$ locus of centroid is parabola

$x = \frac{9}{48} y^{2} + \frac{40}{3}$

$\therefore\; \ell(\text{L.R.}) = \dfrac{48}{9}$

$\therefore\; 3\ell(\text{L.R.}) = \dfrac{48}{3} = 16$

← Back to full paper

Practice more Mathematics

Concept-wise practice with instant solutions on Fundamenthol.

Start practicing →