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|z1 + z2| = |z1| + |z2| is possible if ______. - Mathematics

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MCQ
Fill in the Blanks

|z1 + z2| = |z1| + |z2| is possible if ______.

Options

  • `z_2 = barz_1`

  • `z_2 = 1/z_1`

  • arg (z1) = arg (z2)

  • |z1| = |z2

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Solution

|z1 + z2| = |z1| + |z2| is possible if arg (z1) = arg (z2).

Explanation:

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

Since |z1 + z2| = |z1| + |z2|

 |z1 + z2| = r1 cos θ1 + i r1 sin θ1 + r2 cos θ2 + i r2 sin θ2

|z1 + z2| = `sqrt(r_1^2 cos^2 theta_ + r_2^2 cos^2 theta_2 + 2r_1r_2 cos theta_1 cos theta_2 + r_1^2 sin^2 theta_1 + r_2^2 sin^2 theta_2 + 2r_1r_2 sin theta_1 sin theta_2)`

= `sqrt(r_1^2 + r_2^2 + 2r_1r_2 cos(theta_1 - theta_2))`

But |z1 + z2| = |z1| + |z2|

So `sqrt(r_1^2 + r_2^2 + 2r_1r_2 cos(theta_1 - theta_2))` = r1 + r2

Squaring both sides, we get

`r_1^2 + r_2^2 + 2r_1r_2 cos(theta_1 - theta_2) = r_1^2 + r_2^2 + 2r_1r_2`

⇒ `2r_1r_2 - 2r_1r_2 cos(theta_1 - theta_2)` = 0

⇒ 1 – cos (θ1 – θ2) = 0

⇒ cos (θ1 – θ2) = 1

⇒ θ1 – θ2 = 0

⇒ θ1 = θ2

So, arg (z1) = arg (z2)

Concept: Argand Plane and Polar Representation
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APPEARS IN

NCERT Mathematics Exemplar Class 11
Chapter 5 Complex Numbers and Quadratic Equations
Exercise | Q 47 | Page 96

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