Discussion
A circle with center at \((x, y) = (0.5, 0)\) and radius = 0.5 intersects with another circle with center at \((x, y) = (1, 1)\) and radius = 1 at two points. One of the points of intersection \((x, y)\) is:
Answer: B
One intersection point is (0.2, 0.4).
Write the two circles and subtract one equation from the other to get the line through both intersection points:
\[ \begin{aligned} (x - 0.5)^{2} + y^{2} &= 0.25 \Rightarrow x^{2} - x + y^{2} = 0 \\ (x - 1)^{2} + (y - 1)^{2} &= 1 \\ \Rightarrow x^{2} - 2x + y^{2} - 2y + 1 &= 0 \end{aligned} \]
Subtracting gives \(x + 2y - 1 = 0\), so \(x = 1 - 2y\). Substituting in the first circle: \(5y^{2} - 2y = 0\), so \(y = 0\) or \(y = 0.4\). The points are (1, 0) and (0.2, 0.4).
(0, 0) lies on the first circle but is \(\sqrt{2}\) from (1, 1); (0.5, 0.5) and (1, 2) lie on only one circle or neither.
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