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Parallelogram Condition for a+i, a−i, 1+ai, 1−ai | JEE

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

Question

The complex numbers

a + i

a - i

1 + ai

and

1 - ai

, where

a \in \mathbb{R}

, taken in that order on the Argand plane, represent the vertices of a parallelogram if

a = 1

a = -1

correct

a = 0

Dnone of these

Solution

Step 1: Label the vertices in order:

A = a + i

B = a - i

C = 1 + ai

D = 1 - ai

. Step 2: Use the parallelogram diagonal property. In a parallelogram

ABCD

, the diagonals

AC

and

BD

bisect each other, so their midpoints coincide:

\frac{A + C}{2} = \frac{B + D}{2} \implies A + C = B + D

Step 3: Compute both sides.

A + C = (a + i) + (1 + ai) = (a + 1) + (1 + a)i

B + D = (a - i) + (1 - ai) = (a + 1) - (1 + a)i

Step 4: Set them equal. The real parts

(a + 1)

already match. Equate the imaginary parts:

(1 + a) = -(1 + a) \implies 2(1 + a) = 0 \implies a = -1

Step 5: Check

a = -1

gives genuine (non-degenerate) vertices:

A = -1 + i

B = -1 - i

C = 1 - i

D = 1 + i

— these are four distinct points forming a square (a special parallelogram). Valid. Correct answer: (2)

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