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Two concentric circles C1 and C2 are centered at the origin with radii 4 and 2 respectively. The shaded region is the annular region between them (inside C1 but outside C2). If the point (a, 3) lies inside the shaded region, find the interval of values of a.

1. a > 0
2. −sqrt(7) < a < sqrt(7)
3. a < 3
4. 0 < a < sqrt(7)

Correct answer: −sqrt(7) < a < sqrt(7)

Solution

Point (a, 3) is in the annular shaded region if it is strictly inside C1 (radius 4) and strictly outside C2 (radius 2). The condition inside C1 gives a² + 9 < 16, so a² < 7. The condition outside C2 is automatically satisfied since 3² = 9 > 4 = 2².

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