Maths · Circle, conic sections: A standard form of equations of a circle, the general form of the equation of a circle, its radius and centre
The least area of a circle circumscribing any right triangle of area S is:
The least area of a circle circumscribing any right triangle of area S is:
- A. \( \pi S \)
- B. \( 2 \pi S \)
- C. \( \sqrt{2} \pi S \)
- D. \( 4 \pi S \)
Step-by-step solution
For a right triangle with legs a, b and hypotenuse c, area S = (1/2)ab, and c = √(a²+b²). The circumcircle has diameter c, so its area = π(c/2)² = π(a²+b²)/4. To minimize this area given fixed S, we minimize a²+b² subject to ab = 2S. By AM-GM, a²+b² ≥ 2ab = 4S, with equality when a=b. Thus minimum circumcircle area = π(4S)/4 = πS.
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