# Find the complex number satisfying the equation z+2|(z+1)|+i = 0. - Mathematics

Sum

Find the complex number satisfying the equation z + sqrt(2) |(z + 1)| + i = 0.

#### Solution

Given that: z + sqrt(2) |(z + 1)| + i = 0

Let z = x + yi

∴ (x + yi) + sqrt(2)|(x + yi + 1)| + i = 0

⇒ x + (y + 1)i + sqrt(2)|(x + 1) + yi| = 0

⇒ x + (y + 1)i + sqrt(2) sqrt((x + 1)^2 + y^2) = 0

⇒ x + (y + 1)i + sqrt(2) sqrt(x^2 + 2x + 1 + y^2) = 0 + 0i

⇒ x + sqrt(2) sqrt(x^2 + 2x + 1 + y^2) = 0, y + 1 = 0

⇒ x = - sqrt(2) sqrt(x^2 + 2x + 1 + y^2) and y = –1

⇒ x2 = 2(x2 + 2x + 1 + y2)

⇒ x2 = 2x2 + 4x + 2 + 2y2

⇒ x2 + 4x + 2 + 2y2 = 0

⇒ x2 + 4x + 2 + 2(–1)2 = 0  .....[∵y = –1]

⇒ x2 + 4x + 4 = 0

⇒ (x + 2)2 = 0

⇒ x + 2 = 0

⇒ x = –2

Hence, z = x + yi = –2 – i.

Concept: Algebraic Operations of Complex Numbers
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Chapter 5: Complex Numbers and Quadratic Equations - Exercise [Page 92]

#### APPEARS IN

NCERT Mathematics Exemplar Class 11
Chapter 5 Complex Numbers and Quadratic Equations
Exercise | Q 22 | Page 92

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