$m$ , $n$ and $p$ . Calculate the value of $\dfrac{m + n}{p}$ .

The orange circle and the green circle intersect at the center of the blue circle. If the radius of the orange, green and blue circle are respectively
$4$
$2$
$4\sqrt2$
$8\sqrt2$
$8$
$6$
$6\sqrt2$
$2\sqrt2$

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If we consider the two perpendicular lines as axes, the family of circles that are tangent to both have equations $(x-r)^2+(y-r)^²=r^2$ (the parameter $r$ corresponds to the circles' radii). Which of these pass through the point $(p,p)$ (the centre of the blue circle)?

Plugging in, we have $(p-r)^2+(p-r)^²=r^2$ , or $r^2-4pr+2p^2=0$ .

This is a quadratic in $r$ . The sum of the two solutions is $4p$ (by Vieta).

But clearly the two solutions are $m$ and $n$ . So $m+n=4p$ , and the answer is $\boxed4$