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(D) The point of reflection of $(m, n)$ on the line $y-x=0$ is $(n, m)$.Let the centre of the circle be $(n, m)$ and the radius be $r$.The equation of

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$x^2+y^2-2nx-2my+n^2=0$

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(D) The point of reflection of $(m, n)$ on the line $y-x=0$ is $(n, m)$.Let the centre of the circle be $(n, m)$ and the radius be $r$.The equation of the circle is $(x-n)^2 + (y-m)^2 = r^2$ ... $(i)$.Since the circle touches the $X$-axis,the radius $r$ is equal to the absolute value of the $y$-coordinate of the centre,so $r = |m|$.Thus,$r^2 = m^2$.Substituting $r^2$ into equation $(i)$:$(x-n)^2 + (y-m)^2 = m^2$$x^2 - 2nx + n^2 + y^2 - 2my + m^2 = m^2$$x^2 + y^2 - 2nx - 2my + n^2 = 0$.

$A$ circle is drawn touching the $X$-axis,with its centre at the point of reflection of $(m, n)$ on the line $y-x=0$. Then the equation of the circle is

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