Triangle Inequality
For a complex number z satisfying the constraint |z - 4| ≤ 3, the maximum possible value of the modulus |z| attained on this disk equals which number?
Select the correct option:
Solution
7
The triangle inequality for complex numbers bounds |z| in terms of distances, with |z| = |(z - 4) + 4| ≤ |z - 4| + |4|, a standard JEE Advanced estimation tool. The condition |z - 4| ≤ 3 describes a closed disk of radius 3 centred at the point 4 on the real axis. The farthest point of this disk from the origin lies along the line through the origin and the centre, at distance equal to the centre distance plus the radius, namely 4 + 3 = 7. Thus the maximum of |z| is 7, attained at z = 7. Option 3 reports only the radius. Option 4 reports only the centre distance, ignoring the radius. Option 1 = 4 - 3 gives the minimum of |z|, not the maximum. Hence the maximum modulus is 7. Plausibility check: the point z = 7 satisfies |7 - 4| = 3 ≤ 3 and has |z| = 7, while no point of the disk can exceed centre-distance plus radius, confirming 7 as the extreme value.
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About This Question
- Subject
- mathematics
- Chapter
- complex numbers and quadratic equations
- Topic
- triangle inequality
- Difficulty
- Easy
- Year
- 2025
Solution
Correct Answer:
7
The triangle inequality for complex numbers bounds |z| in terms of distances, with |z| = |(z - 4) + 4| ≤ |z - 4| + |4|, a standard JEE Advanced estimation tool. The condition |z - 4| ≤ 3 describes a closed disk of radius 3 centred at the point 4 on the real axis. The farthest point of this disk from the origin lies along the line through the origin and the centre, at distance equal to the centre distance plus the radius, namely 4 + 3 = 7. Thus the maximum of |z| is 7, attained at z = 7. Option 3 reports only the radius. Option 4 reports only the centre distance, ignoring the radius. Option 1 = 4 - 3 gives the minimum of |z|, not the maximum. Hence the maximum modulus is 7. Plausibility check: the point z = 7 satisfies |7 - 4| = 3 ≤ 3 and has |z| = 7, while no point of the disk can exceed centre-distance plus radius, confirming 7 as the extreme value.
This easy difficulty mathematics question is from the chapter complex numbers and quadratic equations, covering the topic of triangle inequality. It appeared in the 2025 exam.
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