Let the locus of the mid-point of the chord through the origin O of the parabola $y^{2} = 4x$ be the

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Let the locus of the mid-point of the chord through the origin O of the parabola $y^{2} = 4x$ 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 :

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$y^{2} = 4x$Locus of mid point of OP$${M(h,k) \Rightarrow h = \frac{t^{2}}{2},k = t }{\Rightarrow k^{2} = 2	ext{ }h \Rightarrow y^{2} = 2x }$$$$S:y^{2} = 2x $$R (h, k)$${\Rightarrow h = \frac{\frac{3t^{2}}{2}}{4},k = \frac{3t}{4} }{t^{2} = \frac{8	ext{ }h}{3},t = \frac{4k}{3} }{\Rightarrow \frac{16k^{2}}{9} = \frac{8	ext{ }h}{3} \Rightarrow 2k^{2} = 3	ext{ }h }$$Locus of R : $2y^{2} = 3x$

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