complex numbers - Find a connection how the real part of z depends on the imaginary part

Find a connection how the real part of z depends on the imaginary part, if the following two conditions for the complex number z apply:

1. |z|=k, where k is a real number.




2. The real part and the imaginary part of z are positive?

This is what I think:
If the complex number z is z=a+ib then the absolute value is |z|=sqrt(a^2+b^2)=k

If a and b or a or b were negative, the absolute value would still be positive.

Am I anywhere near the answer?

Appreciate your help.

Answer

You're almost there:
\begin{align}
& \sqrt{a^2+b^2} = k \\[10pt]
& a^2+b^2 = k^2 \\[10pt]
& a^2 = k^2 - b^2 \\[10pt]

& a = \sqrt{k^2 - b^2}
\end{align}
and we don't need to say $\text{“}{\pm}\text{''}$ because we know $a\ge0.$