Let be the parabola with its vertex at O. Let P be a point on the parabola and A be a point on the x-axis such that . Then the locus of the centroid of such triangles OPA is

Question

Let $y^{2} = 12 x$ be the parabola with its vertex at O. Let P be a point on the parabola and A be a point on the x-axis such that $∠ O P A = 90 °$ . Then the locus of the centroid of such triangles OPA is : [2026]

Smart Achievers · Accepted Answer

(3)

$m_{A P} = \frac{- t}{2}$

Equation of AP is

$y - 6 t = \frac{- t}{2} (x - 3 t^{2})$

Put $y = 0$

$\Rightarrow x = 12 + 3 t^{2}$

$\Rightarrow A (12 + 3 t^{2}, 0)$

Let centroid of $∆ O P A$ be $G (h, k)$

$\Rightarrow 3 h = 0 + 3 t^{2} + 12 + 3 t^{2}$

$3 k = 0 + 6 t + 0$

$\Rightarrow t = \frac{k}{2}, h = 2 t^{2} + 4$

$\Rightarrow h = 2 (\frac{k^{2}}{4}) + 4$

$\Rightarrow Locus of (h, k) is$

$y^{2} = 2 x - 8$

Solution

Q.
Let $y^{2} = 12 x$ be the parabola with its vertex at O. Let P be a point on the parabola and A be a point on the x-axis such that $∠ O P A = 90 °$ . Then the locus of the centroid of such triangles OPA is : [2026]

1 $y^{2} - 4 x + 8 = 0$

2 $y^{2} - 6 x + 4 = 0$

3 $y^{2} - 2 x + 8 = 0$

4 $y^{2} - 9 x + 6 = 0$

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Solution

Q. Let y2=12x be the parabola with its vertex at O. Let P be a point on the parabola and A be a point on the x-axis such that ∠OPA=90°. Then the locus of the centroid of such triangles OPA is : [2026]

1 y2-4x+8=0

2 y2-6x+4=0

3 y2-2x+8=0

4 y2-9x+6=0