On-the-Argand-Diagram-the-variable-point-Z-represents-a-complex-number-z-Find-the-equation-of-the-locus-of-a-point-Z-which-moves-such-that-z-1-z-2-2-

Question Number 141681 by ZiYangLee last updated on 22/May/21

On the Argand Diagram, the variable point

Z represents a complex number z .

Find the equation of the locus of a point

Z which moves such that ∣ \frac{z - 1}{z + 2} ∣= 2

Answered by MJS_new last updated on 22/May/21

generally

let z = x + y i

∣ \frac{x + y i + a + b i}{x + y i + c + d i} ∣= r; r > 0 \land r \neq 1

leads to

{(x - \frac{a - {c r}^{2}}{r^{2} - 1})}^{2} + {(y - \frac{b - {d r}^{2}}{r^{2} - 1})}^{2} = \frac{({(a - c)}^{2} + {(b - d)}^{2}) r^{2}}{{(r^{2} - 1)}^{2}}

this is a circle with center (\begin{matrix} \frac{a - {c r}^{2}}{r^{2} - 1} \\ \frac{b - {d r}^{2}}{r^{2} - 1} \end{matrix}) and

radius \frac{r}{r^{2} - 1} \sqrt{{(a - c)}^{2} + {(b - d)}^{2}}

for r = 1 we get a line

2 (a - c) x + 2 (b - d) y + a^{2} + b^{2} - c^{2} - d^{2} = 0

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