The locus of the centre of the circle which c | vedclass

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(C) Let the circle be $x^2 + y^2 + 2gx + 2fy + c = 0$. The length of the chord cut by the circle on the $x$-axis $(y=0)$ is $2\sqrt{g^2 - c} = 2a$,whi

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$x^2 + 2by = a^2 - b^2$

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(C) Let the circle be $x^2 + y^2 + 2gx + 2fy + c = 0$. The length of the chord cut by the circle on the $x$-axis $(y=0)$ is $2\sqrt{g^2 - c} = 2a$,which implies $g^2 - c = a^2$,or $c = g^2 - a^2$ $(i)$. The circle passes through the point $(0, b)$ on the $y$-axis,so $0^2 + b^2 + 2g(0) + 2f(b) + c = 0$,which gives $b^2 + 2fb + c = 0$ $(ii)$. Substituting $c = g^2 - a^2$ from $(i)$ into $(ii)$,we get $b^2 + 2fb + g^2 - a^2 = 0$. Rearranging,we have $g^2 + 2fb = a^2 - b^2$. Replacing $(g, f)$ with $(x, y)$ to find the locus,we get $x^2 + 2by = a^2 - b^2$.

The locus of the centre of the circle which cuts a chord of length $2a$ from the positive $x$-axis and passes through a point on the positive $y$-axis at a distance $b$ from the origin is:

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