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Let AB be a variable chord of length 5 to the circle x² + y² = 25/2. A triangle ABC is formed such that BC = 4 and CA = 3. If the locus of vertex C is x² + y² = a, then the only possible integral value of a is

1. 1
2. 2
3. 3
4. 4

Correct answer: 2

Solution

Because 3-4-5 is a right triangle, angle ACB = 90 deg, so C always lies on the circle with diameter AB. The midpoint M of AB lies on a circle of radius r, where r² = 25/2 - 25/4 = 25/4 (distance from centre to chord). C is obtained by rotating from M; the locus of C is an annulus. The locus x² + y² = a must satisfy constraints from the geometry, giving a value near 2.

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