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
Create a free account to view solution
View Solution FreeMore Complex Numbers Questions
The polar form of −5(cos 40o − i sin 40o) is...If a variable circle S touches S1 : |z - z1| = r1 internally and S2 : |z - z2| = r2 externally while the curves S1 & S2 ...z + a + z + b = 0 is the equation of a circle, if -...Which of the following is a complex number...If z and ω are two non-zero complex numbers such that | zω | = 1, and Arg (z) - Arg(ω) = π/2, then &...