Question

**Given:** Parabola $y^2 = 8x$, so $a = 2$ and focus $A = (2, 0)$. Parametric coordinates of points on the parabola: $B = (2t_1^2,\ 4t_1)$ and $C = (2t_2^2,\ 4t_2)$. **Step 1: Use centroid condition.** $$\frac{2 + 2t_1^2 + 2t_2^2}{3} = \frac{7}{3} \implies t_1^2 + t_2^2 = \frac{5}{2}$$ $$\frac{0 + 4t_1 + 4t_2}{3} = \frac{4}{3} \implies t_1 + t_2 = 1$$ **Step 2: Find $t_1 t_2$.** $$(t_1 + t_2)^2 = t_1^2 + t_2^2 + 2t_1 t_2 \implies 1 = \frac{5}{2} + 2t_1 t_2 \implies t_1 t_2 = -\frac{3}{4}$$ **Step 3: Find $(t_1 - t_2)^2$.** $$(t_1 - t_2)^2 = (t_1 + t_2)^2 - 4t_1 t_2 = 1 + 3 = 4$$ **Step 4: Compute $(BC)^2$.** $$ (BC)^2 = (2t_1^2 - 2t_2^2)^2 + (4t_1 - 4t_2)^2 = 4(t_1+t_2)^2(t_1-t_2)^2 + 16(t_1-t_2)^2 $$ $$ = (t_1-t_2)^2\bigl[4(t_1+t_2)^2 + 16\bigr] = 4\times[4\times 1 + 16] = 4 \times 20 = 80 $$ **Answer: (B) $80$**