If z₁, z₂ Are Two Distinct Complex Numbers Such That |z₁−2z₂|/|1/2−z₁z̄₂| = 2, Then

JEE Main 2024 — 6 April, Evening Shift. Previous Year Question.

Problem

If $z_1, z_2$ are two distinct complex numbers such that $\dfrac{|z_1-2z_2|}{\left|\frac{1}{2}-z_1\bar{z_2}\right|}=2$, then:

Key insight. Squaring the given equation and expanding both sides using $|w|^2=w\bar{w}$ produces a mix of terms — but a pair of “cross terms” ($z_1\bar{z_2}$ and $\bar{z_1}z_2$) appear identically on both sides and cancel entirely. What survives factors neatly into a product of two brackets, each depending on only one of $|z_1|$ or $|z_2|$ — turning an “and” condition into an “either/or.”

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Approach

Square both sides of the given equation to remove the moduli, then expand each side using $|w|^2=w\bar w$. Collect like terms — the cross terms involving both $z_1$ and $z_2$ together will cancel — and factor the remaining expression in terms of $|z_1|^2$ and $|z_2|^2$ to see what condition it forces.

Solution

Step 1 — Square both sides

$$|z_1-2z_2|^2 = 4\left|\frac{1}{2}-z_1\bar{z_2}\right|^2$$

Step 2 — Expand the left side using |w|² = w w̄

$$(z_1-2z_2)(\bar{z_1}-2\bar{z_2}) = z_1\bar{z_1} – 2z_1\bar{z_2} – 2z_2\bar{z_1} + 4z_2\bar{z_2}$$

$$= |z_1|^2 + 4|z_2|^2 – 2z_1\bar{z_2} – 2\bar{z_1}z_2$$

Step 3 — Expand the right side

$$4\left(\frac{1}{2}-z_1\bar{z_2}\right)\left(\frac{1}{2}-\bar{z_1}z_2\right) = 4\left[\frac{1}{4} – \frac{1}{2}\bar{z_1}z_2 – \frac{1}{2}z_1\bar{z_2} + z_1\bar{z_2}\bar{z_1}z_2\right]$$

$$= 1 – 2\bar{z_1}z_2 – 2z_1\bar{z_2} + 4|z_1|^2|z_2|^2$$

Step 4 — Set both sides equal and cancel the cross terms

$$|z_1|^2+4|z_2|^2 – 2z_1\bar{z_2}-2\bar{z_1}z_2 = 1-2\bar{z_1}z_2-2z_1\bar{z_2}+4|z_1|^2|z_2|^2$$

The terms $-2z_1\bar{z_2}-2\bar{z_1}z_2$ appear identically on both sides and cancel:

$$|z_1|^2+4|z_2|^2 = 1+4|z_1|^2|z_2|^2$$

Step 5 — Rearrange and factor

Let $x=|z_1|^2$ and $y=|z_2|^2$:

$$x+4y-1-4xy = 0$$

$$(x-1) + 4y(1-x) = 0 \implies (x-1)-4y(x-1) = 0 \implies (x-1)(1-4y) = 0$$

Step 6 — Interpret the factored result

$$x=1 \quad \text{or} \quad y=\frac{1}{4}$$

$$|z_1|^2=1 \quad \text{or} \quad |z_2|^2=\frac{1}{4}$$

$$|z_1|=1 \quad \text{or} \quad |z_2|=\frac{1}{2}$$

Answer

Common mistakes

• Missing that the cross terms cancel identically. Without noticing this, the algebra looks far messier and it’s easy to make a sign error trying to combine four different cross-term expressions instead of two matching pairs.
• Interpreting the factored equation as an “and” instead of an “or.” Since $(x-1)(1-4y)=0$ means at least one factor is zero — not both — the correct interpretation is that either condition alone is sufficient, not that both must hold simultaneously.

Practise next

• If $w_1, w_2$ are distinct complex numbers such that $\dfrac{|w_1-3w_2|}{\left|\frac{1}{3}-w_1\bar{w_2}\right|}=3$, determine the corresponding circle condition on $w_1$ and $w_2$, using the same expand-and-factor approach.