# Toronto Math Forum

## MAT334-2018F => MAT334--Tests => Quiz-1 => Topic started by: Victor Ivrii on September 28, 2018, 04:15:39 PM

Title: Q1: TUT 0301
Post by: Victor Ivrii on September 28, 2018, 04:15:39 PM
$\renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}}$
Describe the locus of points $z$ satisfying the given equation.
\begin{equation*}
|z-1|^2=|z+1|^2+6.
\end{equation*}
Title: Re: Q1: TUT 0301
Post by: Meng Wu on September 28, 2018, 04:34:22 PM
Let $$z=x+iy,\text{where} \space x,y \in \mathbb{R}.$$Given equation becomes:
$$|x+iy-1|^2=|x+iy+1|^2+6$$
Since,$$|x+iy-1|=\sqrt{(x-1)^2+y^2} \\ |x+iy+1|=\sqrt{(x+1)^2+y^2}$$
Hence, we have\require{cancel}\begin{align}\bigg(\sqrt{(x-1)^2+y^2}\bigg)^2&=\bigg(\sqrt{(x+1)^2+y^2}\bigg)^2+6 \\(x-1)^2+y^2&=(x+1)^2+y^2+6 \\ \cancel{x^2}-2x+\cancel{y^2}&=\cancel{x^2}+2x+\cancel{y^2}+6\\-4x&=6\\x&=-\frac{3}{2}\end{align}
$\\$
$\\$
Therefore, the locus of points $z$ are the straight line(vertical line) $x=-\frac{3}{2}.$
Title: Re: Q1: TUT 0301
Post by: Meng Wu on September 28, 2018, 05:51:56 PM
Just notice, I posted the answer before the 18:00 you mentioned. Is this gonna be an issue? Just wondering.