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|（z1+z2)/(z1−z2)|=1 Means z1/z2 Purely Imaginary | JEE

JEE Maths question with a full step-by-step solution.

Question

z_1

and

z_2

are two complex numbers satisfying

\left|\dfrac{z_1 + z_2}{z_1 - z_2}\right| = 1

, then

\dfrac{z_1}{z_2}

is a number which is

Apositive real

Bnegative real

Czero or purely imaginarycorrect

Dnone of these

Solution

Step 1: Clear the modulus. The condition gives

|z_1 + z_2| = |z_1 - z_2|

. Step 2: Divide numerator and denominator inside by

z_2

(assume

z_2 \neq 0

) and let

w = \dfrac{z_1}{z_2}

\left|\frac{z_1 + z_2}{z_1 - z_2}\right| = \left|\frac{w + 1}{w - 1}\right| = 1 \implies |w + 1| = |w - 1|

Step 3: Interpret geometrically.

|w + 1| = |w - 1|

says

w

is equidistant from the points

-1

and

+1

on the real axis. The set of points equidistant from

-1

and

1

is their perpendicular bisector — the imaginary axis. Step 4: Confirm algebraically. Let

w = p + iq

. Then:

|w + 1|^2 = (p + 1)^2 + q^2, \qquad |w - 1|^2 = (p - 1)^2 + q^2

Setting them equal:

(p + 1)^2 = (p - 1)^2 \implies 4p = 0 \implies p = 0

w = iq

is purely imaginary (or zero when

q = 0

). Therefore

\dfrac{z_1}{z_2}

is zero or purely imaginary. Correct answer: (3)

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