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(C) Let the required circle be ${x^2} + {y^2} + 2gx + 2fy + c = 0$ $(i)$.Since it passes through the origin $(0, 0)$,we have $c = 0$.Given that the ce

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

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(C) Let the required circle be ${x^2} + {y^2} + 2gx + 2fy + c = 0$ $(i)$.Since it passes through the origin $(0, 0)$,we have $c = 0$.Given that the centre $(-g, -f)$ lies on $y = x$,we have $-f = -g$,which implies $f = g$.The condition for two circles ${x^2} + {y^2} + 2g_1x + 2f_1y + c_1 = 0$ and ${x^2} + {y^2} + 2g_2x + 2f_2y + c_2 = 0$ to cut orthogonally is $2g_1g_2 + 2f_1f_2 = c_1 + c_2$.Here,$g_1 = g, f_1 = f, c_1 = 0$ and $g_2 = -2, f_2 = -3, c_2 = 10$.Substituting these values,we get $2(g)(-2) + 2(f)(-3) = 0 + 10$.Since $f = g$,this becomes $-4g - 6g = 10$,which gives $-10g = 10$,so $g = -1$.Thus,$f = -1$.The equation of the circle is ${x^2} + {y^2} - 2x - 2y = 0$.

$A$ circle passes through the origin and has its centre on $y = x$. If it cuts ${x^2} + {y^2} - 4x - 6y + 10 = 0$ orthogonally,then the equation of the circle is

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