Exam

JEE Main

Year

2026

Date

28 January

Shift

Evening Shift

Subject

Mathematics

Chapter

Parabola

JEE Main 2026 · 28 January · Evening Shift · Mathematics · Parabola

Let A be the focus of the parabola y^ 2 =8 x . Let the line y=m x+c...

Let A be the focus of the parabola $y^{2}=8 x$. Let the line $y=\mathrm{m} x+\mathrm{c}$ intersect the parabola at two distinct points $B and C$. If the centroid of the triangle $A B C$ is \left(\frac{7}{3}, \frac{4}{3}\right)$, then$(B C)^{2} is equal to :

Step-by-step solution

Coordinates of centroid of triangle ABC are
$\dfrac{2}{3}(t_1^2 + t_2^2 + 1) = \dfrac{7}{3} \Rightarrow t_1^2 + t_2^2 = \dfrac{5}{2}$
$\dfrac{4}{3}(t_1 + t_2) = \dfrac{4}{3} \Rightarrow t_1 + t_2 = 1$
$(t_1 + t_2)^2 = t_1^2 + t_2^2 + 2t_1 t_2 \Rightarrow t_1 t_2 = \dfrac{-3}{4}$
$(t_1 - t_2)^2 = (t_1 + t_2)^2 - 4t_1 t_2 = 4$
$(BC)^2 = 4(t_1^2 - t_2^2)^2 + 16(t_1 - t_2)^2$
$\Rightarrow (BC)^2 = 80$