Parabola
Line Intersecting Parabola — (BC)² from Centroid
nta_pyq_2026_jan
Grade None
Question:
Let $A$ be the focus of the parabola $y^2=8x$. Let the line $y=mx+c$ intersect the parabola at two distinct points $B$ and $C$. If the centroid of the triangle $ABC$ is $\left(\dfrac{7}{3},\dfrac{4}{3}\right)$, then $(BC)^2$ is equal to:
Step-by-Step Solution
Key Concept: $y^2=8x$: $a=2$, focus $A=(2,0)$. Parametric: $B=(2t_1^2,4t_1)$, $C=(2t_2^2,4t_2)$. Centroid $x$: $\tfrac{2}{3}(t_1^2+t_2^2+1)=\tfrac{7}{3}\Rightarrow t_1^2+t_2^2=\tfrac{5}{2}$. Centroid $y$: $\tfrac{4}{3}(t_1+t_2)=\tfrac{4}{3}\Rightarrow t_1+t_2=1$.
$(BC)^2=80$.
Correct Answer: 1