Suppose two circles are centered at \(F_1\) and \(F_2\), respectively, whose centers are at least 2a units apart (see figure). The radius of one circle is 2a + r and the radius of the other circle is r, where \(r\ \geq\ 0\). Show that as r increases, the intersection point P of the two circles describes one branch of a hyperbola with foci at \(F_1\) and \(F_2\).

Solution 75E

Step 1:

Given that

Suppose two circles are centered at F1 and F2, respectively, whose centers are at least 2a units apart (see figure). The radius of one circle is 2a + r and the radius of the other circle is r, where r ≥ 0.