Radius Ratio #2

Without loss of generality let $a = 1$ . then the longest shared chord will be a diameter of this smaller circle.

The respective equations of the two circles are

$(x - 1)^{2} + (y - 1)^{2} = 1$ and $(x - b)^{2} + (y - b)^{2} = b^{2}$ .

By symmetry, the points of intersection of the two circles will lie on the diameter of the smaller circle that has slope $-1$ . Since the diameter passes through $(1,1)$ it will thus lie on the line given by the equation $y - 1 = -(x - 1)$ . Now substitute this into the equation for the smaller circle to find that

$2*(x - 1)^{2} = 1 \Longrightarrow x = 1 \pm \frac{1}{\sqrt{2}}$ .

So one of the points of intersection will then be $(1 - \frac{1}{\sqrt{2}}, 1 + \frac{1}{\sqrt{2}})$ . Now we just need to substitute this point into the equation for the larger circle to find the value for $b$ , (and hence $\frac{b}{a}$ ).

The equation for the larger circle can be written as

$x^{2} + y^{2} - 2b(x + y) + b^{2} = 0$ ,

which, after substituting the coordinates of the point of intersection, yields that

$3 - 4b + b^{2} = 0 \Longrightarrow (b - 3)(b - 1) = 0$ .

As we are looking for a circle with $b \gt (a = 1)$ we can conclude that $b$ , and hence $\frac{b}{a}$ , is equal to $\boxed{3}$ .

√(a^2 + b^2) = √2(b - a)