On this page
Questions like "if |z − 4/z| = 2, find the greatest value of |z|" look as if they need heavy algebra. Most of them need one inequality and a check that the bound can actually be reached. This post explains the triangle inequality for complex numbers, shows where it comes from on the Argand plane, and works through four question types, each with its equality case.
Where is this in the syllabus?
The JEE (Advanced) 2026 brochure lists "polar representation, properties of modulus and principal argument, triangle inequality, cube roots of unity, geometric interpretations." The Syllabus for JEE (Main) 2026 lists the Argand diagram and "modulus and argument (or amplitude) of a complex number".
In school, the picture is narrower this year. The 2026-27 NCERT Class 11 chapter Complex Numbers and Quadratic Equations defines the modulus and shows that it is the distance of z from the origin on the Argand plane, but the inequality itself is not in the chapter. The CBSE 2026-27 mathematics curriculum lists polar representation of complex numbers among Class 11 topics that will be "assessed only formatively". So this is mainly JEE material. Check the documents for your own exam year.
What does the inequality say, and why is it true?
Think of z₁ and z₂ as arrows from the origin. z₁ + z₂ is the third side of a triangle whose other two sides have lengths |z₁| and |z₂|. No side of a triangle is longer than the other two together, and no side is shorter than their difference. So:
||z₁| − |z₂|| ≤ |z₁ + z₂| ≤ |z₁| + |z₂|
Replacing z₂ with −z₂ gives the same bounds for |z₁ − z₂|.
When is it an equality? The right-hand side is reached when the arrows point the same way (one is a positive multiple of the other). The left-hand side is reached when they point in opposite directions. A max-min answer is only correct if you can show the equality case happens.
Worked examples
1. Range of a sum. |z₁| = 6 and |z₂| = 2.5. What values can |z₁ + z₂| take?
From the inequality, 3.5 ≤ |z₁ + z₂| ≤ 8.5. Both ends are reached: 8.5 when z₂ points the same way as z₁, 3.5 when it points the opposite way. Every value in between occurs as z₂ turns.
2. A point on a circle and a fixed point. |z| = 2. Find the greatest and least values of |z + 12 − 5i|.
Let w = 12 − 5i, so |w| = √(144 + 25) = 13. Then 13 − 2 ≤ |z + w| ≤ 13 + 2, so the range is 11 to 15.
Equality check: the maximum needs z to point the same way as w, so z = 2w/13 = (24 − 10i)/13. Then z + w = (180 − 75i)/13, and |z + w| = √(32400 + 5625)/13 = 195/13 = 15. The minimum comes from z = −2w/13.
3. The classic |z − 4/z| = 2 question. Find the greatest and least values of |z|.
Let r = |z| > 0. The left inequality, applied to z and 4/z, gives |r − 4/r| ≤ 2.
If r ≥ 2: r − 4/r ≤ 2, so r² − 2r − 4 ≤ 0, giving r ≤ 1 + √5.
If r < 2: 4/r − r ≤ 2, so r² + 2r − 4 ≥ 0, giving r ≥ √5 − 1.
Equality check: take z real. z = 1 + √5 gives z² − 2z − 4 = 0, so z − 4/z = 2. z = √5 − 1 gives z² + 2z − 4 = 0, so z − 4/z = −2. Both satisfy |z − 4/z| = 2. So the greatest value of |z| is 1 + √5 ≈ 3.24 and the least is √5 − 1 ≈ 1.24. A numerical search over many points on the curve agreed with these bounds.
4. When the inequality becomes a locus. Describe all z with |z − 3| + |z − 3i| = 3√2.
The distance from 3 to 3i is |3 − 3i| = 3√2. By the triangle inequality, |z − 3| + |z − 3i| ≥ 3√2 for every z, with equality only when z lies on the straight segment between the two points. So the locus is that segment, not an ellipse. If the right side had been larger than 3√2, it would have been an ellipse; if smaller, there would be no points at all.
Which mistakes cost the most marks?
Writing |z₁ + z₂| = |z₁| + |z₂|. It is only an inequality, with equality in one special case.
Forgetting the lower bound. ||z₁| − |z₂|| matters just as often in "least value" questions.
Skipping the equality check. A bound you cannot reach is not the maximum.
Dropping the absolute value on the left. |z₁| − |z₂| can be negative; the bound is its size.
Calling every "sum of distances" an ellipse. Compare the constant with the distance between the two fixed points first.
For more Argand-plane geometry, see our post on complex numbers for JEE, and for a related topic, cube roots of unity.
A practice routine
Day 1: Prove both halves of the inequality from the triangle picture, then state each equality case in words.
Day 2: Six "range of |z + w|" questions with |z| fixed. Write the equality case for each end.
Day 3: Three questions like example 3 with different constants, such as |z + 1/z| = 3.
Day 4: Two locus questions like example 4, one where the constant is too small.
If you cannot picture where z must be, the Fermi Simulation Lab has interactive simulations for ideas that are easier to see than to read.
Next time you meet a max-min question on |z|, rewrite it as a distance between two points before you write any algebra.
