Let the locus of the mid-point of the chord through the origin O of the parabola y^2=4 x b…

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 :

Accepted Answer

Let $M({x}_{1},{y}_{1})$ be the locus of the midpoint of the chord of the parabola ${y}^{2}=4x$

herefore  Equation of the chord is ${S}_{1}={S}_{11}\Rightarrow y{y}_{1}-2(x+{x}_{1})={y}_{2}^{1}-4{x}_{1}$

It passes through $0(0,0)$

$	herefore -2{x}_{1}={y}_{2}^{1}-4{x}_{1}$

Locus of M is ${y}^{2}=2x$

Let $P(\frac{1}{2}{t}^{2},t)$ be a point on this locus

$\begin{gathered} Q(x,y)=(\frac{\frac{3{t}^{2}}{2}+0}{4},\frac{3t+0}{4})\Rightarrow (\frac{3{t}^{2}}{8},\frac{3t}{4}) \ \Rightarrow {t}^{2}=\frac{8x}{3},t=\frac{4y}{3}\Rightarrow \frac{16{y}^{2}}{9}=\frac{8x}{3}\Rightarrow 2{y}^{2}=3x \end{gathered}$

Solutions · 1