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Three vertices of a triangle are given as (a cos t, a sin t), (b sin t, -b cos t), and (1, 0), where t is a real parameter. Find the equation of the locus traced by the centroid of this triangle as t varies.

1. (3x + 1)² + (3y)² = a² - b²
2. (3x - 1)² + (3y)² = a² - b²
3. (3x - 1)² + (3y)² = a² + b²
4. (3x + 1)² + (3y)² = a² + b²

Correct answer: (3x - 1)² + (3y)² = a² + b²

Solution

The centroid (x, y) satisfies 3x - 1 = a cos t + b sin t and 3y = a sin t - b cos t. Squaring and adding both equations eliminates t.

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