Complex NumbersHard
Question
If z1 and z2 are two non-zero complex numbers such that |z1 + z2| = |z1| + |z2|, then arg (z1) - arg (z2) is equal to
Options
A.- π
B.
C.0
D.
Solution
Given, |z1 + z2| = |z1| + |z2|
On squaring both sides, we get
|z1|2 + |z2|2 + 2 |z1| |z2| cos (arg z1 - arg z2)
= |z1|2 + |z2|2 + 2 |z1| |z2|
⇒ 2 |z1| |z2| cos (arg z1 - arg z2) = 2 |z1| |z2|
⇒ cos (arg z1 - arg z2) = 1
⇒ arg (z1) - arg (z2) = 0
On squaring both sides, we get
|z1|2 + |z2|2 + 2 |z1| |z2| cos (arg z1 - arg z2)
= |z1|2 + |z2|2 + 2 |z1| |z2|
⇒ 2 |z1| |z2| cos (arg z1 - arg z2) = 2 |z1| |z2|
⇒ cos (arg z1 - arg z2) = 1
⇒ arg (z1) - arg (z2) = 0
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