- #1

NewtonianAlch

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## Homework Statement

If [itex]\frac{z}{z + 3}[/itex] is purely imaginary, show that z lies on a certain circle and find the equation of that circle.

## The Attempt at a Solution

So,

[itex]\frac{z}{z + 3}[/itex] = [itex]\frac{x + iy}{x + iy + 3}[/itex]

Multiplying by the complex conjugate (and simplifying), we get,

[itex]\frac{x^{2} + y^{2} + 3x + 3iy}{x^{2} + y^{2} + 6x + 9}[/itex]

Since we're only interested in the imaginary part here, I take,

[itex]\frac{3iy}{x^{2} + 6x + 9 + y^{2}}[/itex]

I am not too sure what to do from here, also...does "z lies on a certain circle" mean on the boundary line or anywhere in that enclosed zone including the boundary?