Question:
If \( z_{1}=x+iy \), \( z_{2}=a+ib \) and \( x^{2}+y^{2}=a^{2}+b^{2} \), then \( z_{2} = \)
If \( z_{1}=x+iy \), \( z_{2}=a+ib \) and \( x^{2}+y^{2}=a^{2}+b^{2} \), then \( z_{2} = \)
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Euler's and polar forms are highly efficient tools for complex geometry problems. Remember that \( \text{cis}(\theta) = e^{i\theta} \), and any complex number is explicitly specified by its absolute scale (modulus) and angular direction (argument).
Updated On: Jun 8, 2026
- \( |z_{1}|\text{cis}(\tan^{-1}(\frac{b}{a})) \)
- \( |\overline{z_{2}}|z_{1} \)
- \( z_{1}\text{cis}(\tan^{-1}(\frac{y}{x})) \)
- \( z_{1} \)
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The Correct Option is A
Solution and Explanation
Concept:
Any complex number \( z = c + id \) can be represented in its polar form:
\[
z = |z|(\cos \phi + i \sin \phi) = |z|\text{cis}(\phi)
\]
where \( |z| = \sqrt{c^2 + d^2} \) is the modulus and \( \phi = \tan^{-1}\left(\frac{d}{c}\right) \) is the argument of the complex number.
Step 1: Analyzing the given condition \( x^2 + y^2 = a^2 + b^2 \).
The modulus of \( z_1 \) is given by \( |z_1| = \sqrt{x^2 + y^2} \) and the modulus of \( z_2 \) is given by \( |z_2| = \sqrt{a^2 + b^2} \). The given condition states that: \[ x^2 + y^2 = a^2 + b^2 \implies \sqrt{x^2 + y^2} = \sqrt{a^2 + b^2} \implies |z_1| = |z_2| \]
Step 2: Writing \( z_2 \) in polar form.
The argument of \( z_2 = a + ib \) is \( \theta_2 = \tan^{-1}\left(\frac{b}{a}\right) \). Expressing \( z_2 \) in polar form: \[ z_2 = |z_2|\text{cis}(\theta_2) \] Substituting \( |z_2| = |z_1| \) and \( \theta_2 = \tan^{-1}\left(\frac{b}{a}\right) \): \[ z_2 = |z_1|\text{cis}\left(\tan^{-1}\left(\frac{b}{a}\right)\right) \] This precisely matches option (A).
Step 1: Analyzing the given condition \( x^2 + y^2 = a^2 + b^2 \).
The modulus of \( z_1 \) is given by \( |z_1| = \sqrt{x^2 + y^2} \) and the modulus of \( z_2 \) is given by \( |z_2| = \sqrt{a^2 + b^2} \). The given condition states that: \[ x^2 + y^2 = a^2 + b^2 \implies \sqrt{x^2 + y^2} = \sqrt{a^2 + b^2} \implies |z_1| = |z_2| \]
Step 2: Writing \( z_2 \) in polar form.
The argument of \( z_2 = a + ib \) is \( \theta_2 = \tan^{-1}\left(\frac{b}{a}\right) \). Expressing \( z_2 \) in polar form: \[ z_2 = |z_2|\text{cis}(\theta_2) \] Substituting \( |z_2| = |z_1| \) and \( \theta_2 = \tan^{-1}\left(\frac{b}{a}\right) \): \[ z_2 = |z_1|\text{cis}\left(\tan^{-1}\left(\frac{b}{a}\right)\right) \] This precisely matches option (A).
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