if two are two non zero complex numbers such that |z1-z2|=|z1|+|Z2|,then ArgZ1-argZ2 is equal to

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Solution:

The given equation is |z1-z2|=|z1| |Z2|

Step 1: Converting the equation into polar form

Let z1 = r1(cos θ1 + i sin θ1) and z2 = r2(cos θ2 + i sin θ2)

Now, |z1-z2| = |r1(cos θ1 + i sin θ1) - r2(cos θ2 + i sin θ2)|

|z1-z2| = |(r1cos θ1 - r2cos θ2) + i(r1sin θ1 - r2sin θ2)|

|z1-z2| = ((r1cos θ1 - r2cos θ2)^2 + (r1sin θ1 - r2sin θ2)^2)^(1/2)

|z1| |Z2| = r1 r2

Substituting these values in the given equation, we get:

((r1cos θ1 - r2cos θ2)^2 + (r1sin θ1 - r2sin θ2)^2)^(1/2) = r1 r2

Squaring both sides, we get:

(r1cos θ1 - r2cos θ2)^2 + (r1sin θ1 - r2sin θ2)^2 = r1^2 r2^2

Expanding the squares, we get:

r1^2 cos^2 θ1 - 2 r1 r2 cos θ1 cos θ2 + r2^2 cos^2 θ2 + r1^2 sin^2 θ1 - 2 r1 r2 sin θ1 sin θ2 + r2^2 sin^2 θ2 = r1^2 r2^2

r1^2 + r2^2 - 2 r1 r2 cos(θ1 - θ2) = r1^2 r2^2

Dividing both sides by r1^2 r2^2, we get:

(1/r1^2) + (1/r2^2) - 2 cos(θ1 - θ2)/r1 r2 = 1

Let 1/r1^2 = x and 1/r2^2 = y, then the equation becomes:

x + y - 2xy cos(θ1 - θ2) = 1

Step 2: Finding the value of cos(θ1 - θ2)

Multiplying both sides by r1 r2, we get:

r2 x + r1 y - 2 r1 r2 cos(θ1 - θ2) = r1 r2

Substituting the values of x,

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