# Complex numbers, how to describe and sketch a locus?

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• Apr 24th 2011, 02:36 PM
raayhan
Complex numbers, how to describe and sketch a locus?
I need to describe and sketch a locus:

Modulus((z+24)/(z-8i))=3

where do I begin to describe the centre of this locus without converting it into a cartesian equation?

and, how many ways can the cartesian equation be found?
I found (x-3)^2+(y-9)^2=90 by using z=x+yi

if anyone can help it would be much appreciated. Thanks

Ray
• Apr 24th 2011, 02:45 PM
mr fantastic
Quote:

Originally Posted by raayhan
I need to describe and sketch a locus:

Modulus((z+24)/(z-8i))=3

where do I begin to describe the centre of this locus without converting it into a cartesian equation?

and, how many ways can the cartesian equation be found?
I found (x-3)^2+(y-9)^2=90 by using z=x+yi

if anyone can help it would be much appreciated. Thanks

Ray

You have |z + 24| = 3|z - 8i|. The locus is a Circle of Apollonius.

With a bit of effort you can re-arrange the given relation into the form |z - a| = r and hence recognise it as a circle with radius r and centre at z = a. However, I recommend the Cartesian approach of substituting z = x + iy as being more efficient.

I haven't checked your answer (except to check that you get a circle, which you did) - I assume if you can do basic algebra you will get the correct answer.
• Apr 24th 2011, 04:29 PM
Soroban
Hello, raayhan!

Your answer is correct! . . . Good work!

Quote:

I need to describe and sketch a locus: .|(z+24)/(z-8i)| .= .3

Where do I begin to describe the centre of this locus
without converting it into a cartesian equation?
and how many ways can the cartesian equation be found?

I found: .(x - 3)^2 + (y - 9)^2 .= /90 . by using z = x + yi

mr fantastic's description is correct: a Circle of Apollonius.

We have: .|z + 24| .= .3|z - 8i|

Let P be an arbitrary point (x, y).

The equation says:
. . the distance from P to A(-24,0) is three times its distance from B(0, 8).
. . . . . . . . . ._______________ . . . . . ____________
We have: . √(x + 24)^2 + y^2 . = . 3√x^2 + (y-8)^2

. . . . . x^2 + 48x + 576 + y^2 . = . 9(x^2 + y^2 - 16y + 64)

. . . . . x^2 + 48x + 576 + y^2 . = . 9x^2 + 9y^2 - 144y + 576

- - -8x^2 - 48x + 8y^2 - 144y . = . 0

- - . . . . .x^2 - 6x + y^2 - 18y . = . 0

x^2 - 6x + 9 + y^2 - 18y + 81 . = . 0 + 9 + 81

Therefore: . (x - 3)^2 + (y - 9)^2 .= .90

• May 7th 2011, 03:10 PM
raayhan
how do I begin to re-arrange into the form |z - a| = r? where do I begin, what is r and a? thanks
• May 7th 2011, 05:50 PM
mr fantastic
Quote:

Originally Posted by raayhan
how do I begin to re-arrange into the form |z - a| = r? where do I begin, what is r and a? thanks

This way of doing it is more difficult than the straightforward method of substituting z = x + iy. I suggest you stick with that method.