# State True or False for the following: Let z1 and z2 be two complex numbers such that |z1 + z2| = |z1| + |z2|, then arg(z1 – z2) = 0. - Mathematics

MCQ
True or False

State True or False for the following:

Let z1 and z2 be two complex numbers such that |z1 + z2| = |z1| + |z2|, then arg(z1 – z2) = 0.

• True

• False

#### Solution

This statement is True.

Explanation:

Let z1 = x1 + y1i and z2 = x2 + y2i

⇒ |z1 + z2| = |z1| + |z2|

⇒ |x1 + y1i + x2 + y2i| = |x1 + y1i| + |x2 + y2i|

⇒ |(x1 + x2) + (y1 + y2)i| = |(x1 + y1i)| + |(x2 + y2i)|

⇒ sqrt((x_1 + x_2)^2 + (y_1 + y_2)^2) = sqrt(x_1^2 + y_1^2) + sqrt(x_2^2 + y_2^2)

Squaring both sides, we get

⇒ (x_1 + x_2)^2 + (y_1 + y_2)^2 = x_1^2 + y_1^2 + x_2^2 + y_2^2 + 2sqrt((x_1^2 + y_1^2)(x_2^2 + y_2^2))

⇒ x_1^2 + x_2^2 + 2x_1x_2 + y_1^2 + 2y_1y_2 = x_1^2 + y_1^2 + x_2^2 + y_2^2 + 2sqrt(x_1^2x_2^2 + x_1^2y_2^2 + x_2^2y_1^2 + y_1^2y_2^2)

⇒ 2x_1x_2 + 2y_1y_2 = 2sqrt(x_1^2x_2^2 + x_1^2y_2^2 + x_2^2y_1^2 + y_1^2y_2^2)

⇒ x_1x_2 + y_1y_2 = sqrt(x_1^2x_2^2 + x_1^2y_2^2 + x_2^2y_1^2 + y_1^2y_2^2)

Again squares on both sides, we get

x_1^2x_2^2 + y_1^2y_2^2 + 2x_1y_1x_2y_2 = x_1^2x_2^2 + x_1^2y_2^2 + x_2^2y_1^2 + y_1^2y_2^2

⇒ 2x_1y_1x_2y_2 = x_1^2y_2^2 + x_2^2y_1^2

⇒ x_1^2y_2^2 + x_2^2y_1^2 - 2x_1y_1x_2y_2 = 0

⇒ (x_1y_2 - x_2y_2)^2 = 0

⇒ x_1y_2 - x_2y_1 = 0

⇒ x_1y_2 = x_2y_1

⇒ x_1/y_1 = x_2/y_2

⇒ y_1/x_1 = y_2/x_2

⇒ arg (z1) = arg (z2)

⇒ arg (z1) – arg (z2) = 0

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 26.(vii) | Page 93

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