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Complex Numbers (1 Viewer)

wagig

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Hey guys, i have an exam coming up this thursday and i'm not sure about an answer to a questions:
"On the Argand diagram, clearly illustrate the locus of z if w = (z+2)/(z-4i) is purely imaginary"
 

rumbleroar

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First, I would let z=x+iy
Then I would solve w (realise your fraction,etc)
Then let re(w)=0 since it's purely imaginary
Would probably get a locus and sketch

Hoped it helped!


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SpiralFlex

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You're missing the "y ≠ 0" condition. If Re(w) = 0, y = 0, then it is Real not Imaginary. 0 + 0i = Real.
That's a condition. I was talking about another condition.

The condition was when z = 0 + 4i. i.e (to draw a 'hole' at (0, 4), since it is not part of the locus of z).
 
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ChillTime

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This unassuming question is turning out quite interesting.

You're missing the "y ≠ 0" condition. If Re(w) = 0, y = 0, then it is Real not Imaginary. 0 + 0i = Real.
Did you mean there was a discontinuity at y = 0 or Im(w) = Re(w) = 0?

Because w = that huge expression in my working above.

That's a condition. I was talking about another condition.

The condition was when z = 0 + 4i. i.e (to draw a 'hole' at (0, 4), since it is not part of the locus of z).
Right you are.
 
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anomalousdecay

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That's a condition. I was talking about another condition.

The condition was when z = 0 + 4i. i.e (to draw a 'hole' at (0, 4), since it is not part of the locus of z).
Yes this. Remember that the denominator is NOT equal to zero, hence the condition.
 

ilikecake

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An easier way would be to do this geometrically.

Because the result is purely imaginary, we can simply say that arg[(z+2)/(z-4i)] = +/- (pi/2).
Subsequently, arg (z+2) - arg (z-4i) = +/- (pi/2).
The shape itself becomes a circle, with z+2 and z-4i as two points on the circle that, when you draw a line from each to another point on the circle, the angle formed is pi/2.
 

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