Talk:Equation of Circle in Complex Plane/Examples/z (conj z + 2) = 3

Because it doesn't describe a circle.

\(\ds z \paren {\overline z + 2}\)

\(=\)

\(\ds 3\)

\(\ds \leadsto \ \ \)

\(\ds \size z^2 + 2 z\)

\(=\)

\(\ds 3\)

$\implies z = \dfrac {\size z^2 - 3} 2 \in \R$

\(\ds \leadsto \ \ \)

\(\ds z^2 + 2 z - 3\)

\(=\)

\(\ds 0\)

\(\ds \leadsto \ \ \)

\(\ds z\)

\(=\)

\(\ds 1 \text{ or } -3\)

RandomUndergrad (talk) 03:11, 22 April 2020 (EDT)

To clarify, $\paren {x + 1}^2 + \paren {y + i}^2 = 3$ doesn't describe a circle, since $y$ is real.

If it does, the circle would have center $\tuple {-1, -i} \notin \R^2$. RandomUndergrad (talk) 03:15, 22 April 2020 (EDT)

Hmm ... wonder what was going through Spiegel's mind when he wrote that? From the book:

$71.$ Describe and graph the locus represented by each of the following: ... (d) $z \paren {\overline z + 2} = 3$ ... Ans. ... (d) circle, ...

I think it needs an entry in the errata section. We remember how we were warned about the trustworthiness and accuracy of Schaum manuals at Uni. This is perhaps one reason why. --prime mover (talk) 03:57, 22 April 2020 (EDT)

$\implies z = \dfrac {3 - \size z^2} 2 \in \R$, surely? --prime mover (talk) 03:59, 22 April 2020 (EDT)

Definition:Complex Modulus: The complex modulus is a real-valued function. RandomUndergrad (talk) 09:38, 22 April 2020 (EDT)

But $\size z^2 + 2 z = 3 \leadsto 3 - \size z^2 = 2 z \leadsto z = \dfrac {3 - \size z^2} 2$, yes? Although of course $\dfrac {3 - \size z^2} 2 = -\dfrac {\size z^2 - 3} 2$ so yes, they are both real. I just wondered why the order of terms on the numerator were changed. --prime mover (talk) 09:51, 22 April 2020 (EDT)

Right. I see that now. RandomUndergrad (talk) 10:30, 22 April 2020 (EDT)

The correction has now been published, as has the addition to the errata page. --prime mover (talk) 10:10, 22 April 2020 (EDT)