Value of Radius Square for Circle Given Common Tangents Midpoints Line

Let $C_1$ be the circle defined by $x^2 + y^2 = 1$ with center at the origin $O(0,0)$ and radius $R_1 = 1$.

Let $C_2$ be the circle defined by $(x - 4)^2 + (y - 1)^2 = r^2$ with center $A(4,1)$ and radius $R_2 = r$, where $1 < r < 3$.

Expanding the equation of $C_2$, we get: $C_2: x^2 + y^2 - 8x - 2y + 17 - r^2 = 0$

Let $M$ be the midpoint of the common tangent segment $PQ$, where $P$ lies on $C_1$ and $Q$ lies on $C_2$. By definition, $M$ is equidistant from the points of contact $P$ and $Q$, which means $MP = MQ$.

The power of point $M$ with respect to $C_1$ is $(MP)^2$, and the power of point $M$ with respect to $C_2$ is $(MQ)^2$. Since $MP = MQ$, the power of $M$ with respect to both circles is equal: $\text{Power}(M, C_1) = \text{Power}(M, C_2)$

Hence, the midpoint $M$ lies on the radical axis of $C_1$ and $C_2$. By a similar argument, the midpoint of the second common tangent segment $ST$ also lies on the radical axis of the two circles.

Therefore, the line joining the midpoints of $PQ$ and $ST$ is precisely the radical axis of $C_1$ and $C_2$.

The equation of the radical axis is given by $C_1 - C_2 = 0$: $(x^2 + y^2 - 1) - (x^2 + y^2 - 8x - 2y + 17 - r^2) = 0$ $8x + 2y - 18 + r^2 = 0$

The line meets the $x$-axis at point $B$. To find the coordinates of $B$, we set $y = 0$: $8x - 18 + r^2 = 0 \implies x = \frac{18 - r^2}{8}$

So, the coordinates of $B$ are $B\left(\frac{18 - r^2}{8}, 0\right)$.

We are given that $A = (4,1)$ and the distance $AB = \sqrt{5}$, which implies $AB^2 = 5$. Using the distance formula: $AB^2 = \left(4 - \frac{18 - r^2}{8}\right)^2 + (1 - 0)^2 = 5$

Simplifying the expression inside the brackets: $\left(\frac{32 - 18 + r^2}{8}\right)^2 + 1 = 5$ $\left(\frac{14 + r^2}{8}\right)^2 = 4$

Taking the square root on both sides: $\frac{14 + r^2}{8} = 2 \quad \text{or} \quad \frac{14 + r^2}{8} = -2$

Since $r$ is a real number, $14 + r^2 > 0$, so we take the positive value: $14 + r^2 = 16 \implies r^2 = 2$

Since $1 < r < 3$, $r = \sqrt{2} \approx 1.414$ satisfies the given range.

Thus, the value of $r^2$ is 2.