Consider the following

1. zz̅ = |z|²

2. z^-1 = \(\rm \frac {z}{|z|^2}\), where z = complex number

Which of the above statement is/are correct?

Question

Question

Answer 1

Concept:

Consider a complex number, z = a + ib

Conjugate of complex number = z̅ = a - ib

Modulus of complex number = |z| = \(\rm \sqrt{(a^2 + b^2) }\)

Calculation:

Let, z = a + ib,

zz̅ = (a + ib)(a - ib)

= \(\rm a^2-(ib)^2\)

= \(\rm a^2-i^2(b)^2\)

=\(\rm a^2-(ib)^2\)

=\(\rm a^2+b^2\cdots (\because i^2=-1)\)

And, |z|² = \(\rm (\sqrt{a^2+b^2})^2\)

= \(\rm {a^2+b^2}\)

∴ zz̅ = |z|²

Now,

\(\rm z^{-1}=\frac1 z=\frac{1}{a+ib}\)

=\(\rm \frac{1}{a+ib}\times \frac{a-ib}{a-ib}=\frac{a-ib}{a^2+b^2}\)

= \(\rm \frac{\bar z}{|z|^2}\) ≠ \(\rm \frac {z}{|z|^2}\)

Hence, option (1) is correct.