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An equilateral triangle has one of its vertices at the origin and the other two vertices on the parabola y² = 16x. What is the side length of this triangle?
- 4*sqrt(3)
- 16*sqrt(3)
- 32*sqrt(3)
- 24*sqrt(3)
Correct answer: 32*sqrt(3)
Solution
The two non-origin vertices are symmetric about the x-axis: P = (h, k) and Q = (h, -k) on y² = 16x, so k² = 16h. The equilateral condition OP = PQ gives sqrt(h² + k²) = 2k, hence h = k*sqrt(3). Substituting into k² = 16h yields k = 16*sqrt(3) and side = 2k = 32*sqrt(3).
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