The locus of the midpoints of the chords of t | vedclass

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(A) The equation of the chord of a circle $S \equiv x^2+y^2+2gx+2fy+c=0$ with a given midpoint $M(x_1, y_1)$ is given by $T=S_1$,which is $x x_1 + y y

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$x^2+y^2-x=0$

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(A) The equation of the chord of a circle $S \equiv x^2+y^2+2gx+2fy+c=0$ with a given midpoint $M(x_1, y_1)$ is given by $T=S_1$,which is $x x_1 + y y_1 + g(x+x_1) + f(y+y_1) + c = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c$. Given the circle $x^2+y^2-2x=0$,we have $g=-1, f=0, c=0$. The chord passes through the point $(0,0)$. Substituting $(0,0)$ into the equation $T=S_1$: $0(x_1) + 0(y_1) - 1(0+x_1) + 0(0+y_1) + 0 = x_1^2 + y_1^2 - 2x_1$. $-x_1 = x_1^2 + y_1^2 - 2x_1$. $x_1^2 + y_1^2 - x_1 = 0$. Replacing $(x_1, y_1)$ with $(x, y)$,the locus is $x^2+y^2-x=0$.

The locus of the midpoints of the chords of the circle $x^2-2x+y^2=0$ drawn from the point $(0,0)$ on it is

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