What is the radius of S?

For the following two (02) items: Suppose S is the sphere with the <strong>SMALLEST</strong> radius that passes through the points A(1,0,0), B(0,1,0) and C(0,0,1).

  1. A. \sqrt{\frac{1}{3}}
  2. B. \sqrt{\frac{2}{3}} ✓
  3. C. \frac{1}{3}
  4. D. 1

Correct Answer: B. \sqrt{\frac{2}{3}}

Explanation

Let the center of the sphere be (x, y, z). Since it is equidistant from A, B, C, we have x=y=z. The squared radius is r^2 = (x-1)^2 + x^2 + x^2 = 3x^2 - 2x + 1. To minimize r^2, take the derivative with respect to x and set to zero: 6x - 2 = 0 \implies x = 1/3. The minimum r^2 is 3(1/3)^2 - 2(1/3) + 1 = 2/3. The radius is \sqrt{2/3}.

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