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# Modulus of complex numbers (FP2, OCR) watch

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1. Hi guys, I'm currently stuck on a question about modulus; I have to identify the points corresponding to equations on an argand diagram, and I've got stuck on this one:

'|z + 4| = 3|z|' - I tried to work it out algebraically (I put z = x + yi), squared both sides, turned the left hand into (z + 4)(z* + 4), got zz* + 4(z* + z) + 16 out of it; I subtracted a zz* = |z|^2 (so on the other side there is now 8|z|^2 rather than 9). I equated z* + z to 2x, multiplied it by 4, and ended up with 8x + 16 = 8|z|^2 . I then divided by 8, got x + 2 = |z|^2 , and I'm now stuck...

What I'm looking for is apparently a circle of centre 1/2 + 0i and radius 1.5, but I don't see where what comes from... can anyone help me?
2. How can you write in terms of x and y? Do that, then you have something that might look familiar.
3. No need to get mucky with complex conjugates.

|(x+4)+iy|^2 = 9|x+iy|^2

so (x+4)^2 + y^2 = 9x^2 + 9y^2

Now rearrange, complete the square on the 8x^2 - 8x term, and you should have the standard equation of a circle in Cartesian coordinates, with the centre and radius easy to read off.
4. I have to say, I have no idea what you just did there Could you explain what you did?
5. (Original post by Kyta)
I have to say, I have no idea what you just did there Could you explain what you did?
Glutamic Acid used the method that I'd probably use in doing this; that is, start by writing , square both sides and rearrange to get the Cartesian equation of a circle. However, his method started from the beginning, which you don't need to do -- you can finish off the question from the point you got stuck in your post, by writing and rearranging from there.
6. Thanks a lot; I ended up getting the right answer. The fact that |z| = sqrt(x^2 + y^2) slipped my mind near the end there, lol. Thanks a lot to both of you

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