Complex Numbers (1 Viewer)

[wagig]

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]

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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[ChillTime]

 

[SpiralFlex]

Aren't we forgetting something important guys?

 

[awper7]

Aren't we forgetting something important guys?

You're missing the "y ≠ 0" condition. If Re(w) = 0, y = 0, then it is Real not Imaginary. 0 + 0i = Real.

 

[SpiralFlex]

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).

 

[ChillTime]

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.

 

[anomalousdecay]

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.

 

[wagig]

Wow thanks everyone

 

[ilikecake]

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.