Let the locus of the mid-point of the chord through the origin O of the parabola be the curve S. Let P be any point on S. Then the locus of the point, which internally divides OP in the ra

Question

Let the locus of the mid-point of the chord through the origin O of the parabola $y^{2} = 4 x$ be the curve S. Let P be any point on S. Then the locus of the point, which internally divides OP in the ratio 3:1, is: [2026]

Smart Achievers · Accepted Answer

(2)

$y^{2} = 4 x$

$Locus of mid point of O P$

$M (h, k) \Rightarrow h = \frac{t^{2}}{2}, k = t$

$\Rightarrow k^{2} = 2 h \Rightarrow y^{2} = 2 x$

$S : y^{2} = 2 x$

$\Rightarrow h = \frac{\frac{3 t^{2}}{2}}{4}, k = \frac{3 t}{4}$

$t^{2} = \frac{8 h}{3}, t = \frac{4 k}{3}$

$\Rightarrow \frac{16 k^{2}}{9} = \frac{8 h}{3}$ $\Rightarrow 2 k^{2} = 3 h$

$Locus of R : 2 y^{2} = 3 x$

Solution

Q.
Let the locus of the mid-point of the chord through the origin O of the parabola $y^{2} = 4 x$ be the curve S. Let P be any point on S. Then the locus of the point, which internally divides OP in the ratio 3:1, is: [2026]

1 $3 x^{2} = 2 y$

2 $2 y^{2} = 3 x$

3 $2 x^{2} = 3 y$

4 $3 y^{2} = 2 x$

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Solution

Q. Let the locus of the mid-point of the chord through the origin O of the parabola y2=4x be the curve S. Let P be any point on S. Then the locus of the point, which internally divides OP in the ratio 3:1, is: [2026]

1 3x2=2y

2 2y2=3x

3 2x2=3y

4 3y2=2x