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Q.

Let R be the focus of the parabola $y^{2} = 20 x$ and the line y = mx + c intersect the parabola at two points P and Q. Let the point G(10, 10) be the centroid of the triangle PQR. If c – m = 6, then ${(PQ)}^{2}$ is

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a

346

b

296

c

325

d

317

answer is B.

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Detailed Solution

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$\begin{array}{l} \begin{array}{l} \frac{5 (t_{4}^{2} + t_{2}^{2} + 1)}{3} = 10 \\ ∴ t_{1}^{2} + t_{2}^{2} = 5 \\ \frac{10 (t_{1} + t_{2})}{3} = 10 \\ ∴ t_{1} + t_{2} = 3 \end{array} \end{array}$
$\begin{array}{l} t_{1} = 1, t_{2} = 2 \\ P \equiv (5, 10) Q \equiv (20, 20) \end{array}$
$∴ Equation of P Q = y - 10 = \frac{10}{15} (x - 5)$
$\begin{array}{c} 3 y - 30 = 2 x - 10 \\ y = \frac{2}{3} x + \frac{20}{3} \\ P Q^{2} = 225 + 100 = 325 \end{array}$

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Let R be the focus of the parabola y2=20x and the line y = mx + c intersect the parabola at two points P and Q. Let the point G(10, 10) be the centroid of the triangle PQR. If c – m = 6, then (PQ)2 is