Question

From the origin, chords are drawn to the circle $(x - 1)^2 + y^2 = 1$, then equation of locus of middle points of these chords, is -

• A. $x^2 + y^2 = 1$
• B. $x^2 + y^2 = x$
• C. $x^2 + y^2 = y$
• D. None of these

Answer 1

Answer: (B)

$x^2 + y^2 = x$

Answer 2

Here equation of the given circle is $x^2 + y^2 - 2x = 0$ This clearly passes through origin Hence if $(x_1, \,y_1)$ be midpoint of the chord then its equation is given by $T = S_1$ $\Rightarrow\quad xx_{1} + yy_{1} - \left(x + x_{1}\right) = x_{1}^{2} + y_{1}^{2} - 2x_{1}$ or $\quad xx_{1} + yy_{1} - x = x_{1}^{2} + y_{1}^{2} - x_{1}$ This passes through the origin $\left(0, \,0\right)$ $\therefore\quad x_{1}^{2} + y_{1}^{2} - x_{1} = 0$ $\therefore\quad$ Required locus is $x^{2} + y^{2} = x$