Simply Complex!

Algebra Level 4

A function $f$ is defined by $f(z)=i\overline{z}$ . The value of $z$ which satisfies the condition $|z|=5$ and $f(z)=z$ can be written in the form $a+bi$ . What is the value of $\frac{ab}{2}$ ?

(Answer might have numbers after the decimal point. Don't round off!!)

The answer is 6.25.

1 solution

Rajdeep Ghosh
May 22, 2018

Let $z=a+bi$ . Note that $i\overline{z}=b+ai$ .

According to the second condition- $a+bi=b+ai$ , which implies that a=b.

The first condition gives $\sqrt{a^2+a^2}=5$ .

This gives us - $a=\frac{5}{\sqrt{2}}$ . So the value of $z$ satisfying both conditions is $\frac{5}{\sqrt{2}}+\frac{5}{\sqrt{2}}i$ . So $\frac{ab}{2}=25/4=\boxed{6.25}$

a=-5/sqrt(2) gives a second solution for z. However the final answer is the same.

Joe Mansley - 2 years, 9 months ago

Simply Complex!

The answer is 6.25.

1 solution

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