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Complex Numbers and Quadratic Equations — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSComplex Numbers and Quadratic Equations5 Marks
Question
Solve the equation |z| = z + 1 + 2i.
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Answer

Given that, |z| = z + 1 + 2i
|z| = (z + 1) + 2i
Squaring both sides
|z|2 = |z + 1|2 + 4i2 + 4(z + 1)i
⇒ |z|2 = |z|2 + 1 + 2z - 4 + 4(z + 1)i
⇒ 0 = -3 + 2z + 4(z + 1)i
⇒ 3 - 2z - 4(z + 1)i = 0
⇒ 3 - 2(x + yi) - 4[x + yi + 1]i = 0
⇒ 3 - 2x - 2yi - 4xi - 4yi2 - 4i = 0
⇒ 3 - 2x + 4y - 2yi - 4i - 4xi = 0
⇒ (3 - 2x + 4y) - i(2y + 4x + 4) = 0
⇒ 3 - 2x + 4y = 0
⇒ 2x - 4y = 3
And 4x + 2y + 4 = 0
⇒ 2x + y = -2
Solving eq. (i) and (ii), we get
$\text{y}=-1$ and $\text{x}=-\frac{1}{2}$
Hence, the value of $\text{z}=\text{x}+\text{yi}=\Big(-\frac{1}{2}-\text{i}\Big)$ 

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