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Question

Question: If z is a complex number such that \(A^{2} + B^{2}\) is purely imaginary, then....

If z is a complex number such that A2+B2A^{2} + B^{2} is purely imaginary, then.

A

A2−B2A^{2} - B^{2}

B

A2A^{2}

C

B2B^{2}

D

z2+zˉ=0z^{2} + \bar{z} = 0

Answer

A2A^{2}

Explanation

Solution

Let ∣z2∣=∣(x+iy)2∣|z^{2}| = |(x + iy)^{2}|where =∣x2−y2+2ixy∣=(x2−y2)2+(2xy)2= |x^{2} - y^{2} + 2ixy| = \sqrt{(x^{2} - y^{2})^{2} + (2xy)^{2}}1

This gives =(x2+y2)2= \sqrt{\left( x^{2} + y^{2} \right)^{2}}

=∣z∣2=∣x+iy∣2=(x2+y2)2=x2+y2= |z|^{2} = |x + iy|^{2} = \sqrt{(x^{2} + y^{2})^{2}} = x^{2} + y^{2} ∣z2∣=∣z∣2|z^{2}| = |z|^{2}.