Locus of Complex Numbers

The locus of Complex Numbers is obtained by letting ( z = x+yi ) and simplifying the expressions. Operations of modulus, conjugate pairs and arguments are used to determine the locus of complex numbers.


Let \( z = x + iy \) be a complex number such that \(|z-1|^2-|z-3|^2 = 1 \). Find the locus of all complex numbers \(z\).

\( \begin{aligned} \displaystyle \require{color}
|x+yi-1|^2-|x+yi-3|^2 &= 1 \\
\Big[(x-1)^2 + y^2\Big]-\Big[(x-3)^2 + y^2\Big] &= 1 \\
x^2-2x + 1 + y^2-x^2 + 6x-9-y^2 &= 1 \\
4x &= 9 \\
\therefore x &= \frac{9}{4}
\end{aligned} \)

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  1. Muhammad numan

    There is a mistake in the last line of the answer for the locus of the complex numbers. Instead of 4/9, it should be 9/4.

    1. iitutor Post author

      Thanks for reaching out. It is now rectified and thanks again.

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