From the origin chords are drawn to the circle $(x-1)^{2}+y^{2}=1$. The equation of the locus of the

Question

From the origin chords are drawn to the circle $(x-1)^{2}+y^{2}=1$. The equation of the locus of the midpoints of these chords is

Professor PC Thomas Classes · Answer

Let $P(h, k)$ be the midpoint of a chord $O A$ of the circle $x^{2}+y^{2}-2 x=0$\ $	herefore \quad$ Equation of the chord $O A$ is\ \ $T=S_{1}$\ $\Rightarrow \quad x h+y k-(x+h)=h^{2}+k^{2}-2 h$\ This chord passes through the origin\ $\Rightarrow \quad-h=h^{2}+k^{2}-2 h \Rightarrow h^{2}+k^{2}-h=0$\ $	herefore \quad$ The locus of $P(h, k)$ is $x^{2}+y^{2}-x=0$\