Simple Circle Problem

Question

An elegant circle problem. It goes by many names. This is my version.

1. Dog 1 is tied to a post by a leash 1 unit long.
2. He shares half of his land with Dog 2 tied to a post 1 unit away from his own.
3. How long is Dog 2's leash?

Answer 1

An idea with analytic geometry:

We can assume the first dog's land is the canonical unit circle $\;S:\;x^2+y^2=1\;$ , and then the second dog's post is at $\;(0,1)\;$ , say.

The question can then be rephrased as: what has to be the radius $\;R\;$ of the circle $\;T:\;x^2+(y-1)^2=R^2\;$ so that $\;Area(S)\cap Area(T)=\frac\pi2\;$ ?

Hint:

Using integration and symmetry, we have that

$$\frac\pi4=2\int\limits_0^{x_0}\left[\sqrt{1-x^2}-\left(1+\sqrt{R^2-x^2}\right)\right]dx$$

with $\;x_0\;$ the abscissa of the intersection point of both circles in the first quadrant.

Clearly, $\;0<R\le 2\;$ (why?)