Let A be the focus of the parabola Let the line y=mx+c intersect the parabola at two distinct points B and C. If the centroid of the triangle ABC is , then is equa

Question

Let A be the focus of the parabola $y^{2} = 8 x .$ Let the line y=mx+c intersect the parabola at two distinct points B and C. If the centroid of the triangle ABC is $(\frac{7}{3}, \frac{4}{3})$ , then ${(B C)}^{2}$ is equal to : [2026]

Smart Achievers · Accepted Answer

(3)

$Coordinates of centroid of triangle ABC are$

$\frac{2}{3} (t_{1}^{2} + t_{2}^{2} + 1) = \frac{7}{3} \Rightarrow t_{1}^{2} + t_{2}^{2} = \frac{5}{2}$

$\frac{4}{3} (t_{1} + t_{2}) = \frac{4}{3} \Rightarrow t_{1} + t_{2} = 1$

${(t_{1} + t_{2})}^{2} = t_{1}^{2} + t_{2}^{2} + 2 t_{1} t_{2} \Rightarrow t_{1} t_{2} = - \frac{3}{4}$

${(t_{1} - t_{2})}^{2} = {(t_{1} + t_{2})}^{2} - 4 t_{1} t_{2} = 4$

${(B C)}^{2} = 4 {(t_{1}^{2} - t_{2}^{2})}^{2} + 16 {(t_{1} - t_{2})}^{2}$

$\Rightarrow {(B C)}^{2} = 80$

Solution

Q.
Let A be the focus of the parabola $y^{2} = 8 x .$ Let the line y=mx+c intersect the parabola at two distinct points B and C. If the centroid of the triangle ABC is $(\frac{7}{3}, \frac{4}{3})$ , then ${(B C)}^{2}$ is equal to : [2026]

1 41

2 32

3 80

4 89

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