The locus of the middle point of the chord of the circle $\mathrm{x}^{2}+\mathrm{y}^{2}-2\mathrm{x}=0$ passing through the origin is
- $\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{x}=0$
- $\mathrm{x}^{2}+\mathrm{y}^{2}-\mathrm{y}=0$
- $\mathrm{x}^{2}+\mathrm{y}^{2}-\mathrm{x}=0$
- $\mathrm{x}^{2}+\mathrm{y}^{2}-\mathrm{x}-\mathrm{y}=0$
Let the midpoint be (h, k). The chord passes through the origin (0,0) and (h,k), so its equation is y = (k/h)x. The midpoint of a chord with midpoint (h,k) in a circle x^2 + y^2 - 2x = 0 is given by T = S1, resulting in xh + yk - (x+h) = h^2 + k^2 - 2h. Substituting (0,0) into this satisfies the condition, leading to the locus x^2 + y^2 - x = 0.
Let the midpoint of a chord passing through the origin be (h, k). Using the midpoint chord equation for the circle x^2 + y^2 - 2x = 0, the equation of the chord is T = S1, giving h*x + k*y - (h + x) = h^2 + k^2 - 2h. Rearranging this gives x*(h - 1) + k*y = h^2 + k^2 - h. Since this chord passes through the origin (0, 0), substituting x = 0 and y = 0 yields h^2 + k^2 - h = 0. Replacing h with x and k with y gives the required locus as x^2 + y^2 - x = 0.