JEE Main · MathematicsMedium

$If z_{1} and z_{2} are two complex numbers such that ∣ z_{1} ∣ = 1 and ∣ z_{2} ∣ = 1, then the value of ∣ z_{1} + z_{2} ∣^{2} + ∣ z_{1} - z_{2} ∣^{2} is$

A. $1$
B. $2$
C. $3$
D. $4$

Show correct answer & step-by-step solution

Correct answer: D — $4$

Solution

$Recall the identity ∣ z_{1} + z_{2} ∣^{2} = ∣ z_{1} ∣^{2} + ∣ z_{2} ∣^{2} + 2 Re (z_{1} \overset{z}{ˉ}_{2}) .$
$Recall the identity ∣ z_{1} - z_{2} ∣^{2} = ∣ z_{1} ∣^{2} + ∣ z_{2} ∣^{2} - 2 Re (z_{1} \overset{z}{ˉ}_{2}) .$
$Adding these two equations gives ∣ z_{1} + z_{2} ∣^{2} + ∣ z_{1} - z_{2} ∣^{2} = 2 (∣ z_{1} ∣^{2} + ∣ z_{2} ∣^{2}) .$
$Since ∣ z_{1} ∣ = 1 and ∣ z_{2} ∣ = 1, the sum is $2(1 + 1) = 4$.$
$Hence the answer is (D).$

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