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# loci in complex plane watch

1. not my favourite topic...

It's Q34 Edexcel in Review Exercise, don't know how to rearrange to get the 2nd part of the question. Any help very gratefully accepted

"A transformation from the z plane to the w plane is given by :

w = (z-i)/z

Show that under this transformation the line Im(z) =1/2 is mapped to the circle with equation |w| =1.
Hence or otherwise find in the form w= (az+b)/(cz+d) the transformation that maps the line Im(z) =1/2 to the circle with centre (3-i) and radius 2.
2. (Original post by oceane)
not my favourite topic...

It's Q34 Edexcel in Review Exercise, don't know how to rearrange to get the 2nd part of the question. Any help very gratefully accepted

"A transformation from the z plane to the w plane is given by :

w = (z-i)/z

Show that under this transformation the line Im(z) =1/2 is mapped to the circle with equation |w| =1.
Hence or otherwise find in the form w= (az+b)/(cz+d) the transformation that maps the line Im(z) =1/2 to the circle with centre (3-i) and radius 2.
Let z = x + iy. Then the line Im(z) = 0.5 means z = x + 0.5i (i.e. imaginary part of z is always a half). Therefore:

w = (x + 0.5i - i)/(x + 0.5i)

w = (x - 0.5i)/(x + 0.5i)

Mulitply top and bottom by (x - 0.5i) u get:

w = (x^2 - 1/4 - xi) / (x^2 + 1/4)

If you let w = u + iv then equating real and imaginary coefficients

u = (x^2 - 1/4)/ (x^2 + 1/4) and v = -x/(x^2 + 1\4)

If you do u ^2 + v^2 you will find that all the x's cancel out and the answer is just 1. Therefore the line Im(z) = 0.5 is mapped onto a circle, radius 1, centre (0, 0), i.e. |w| = 1 (same thing).

For the second part, you want to double (enlarge) the radius, and then shift the centre of the circle by 3 - i

Therefore:

double:

w = 2(z - i) / z = (2z - 2i)/z

Then shift centre by adding 3 - i = (3z - iz)/z

w = (2z - 2i)/z + (3z - iz)/z = {(5 - i)z - 2i}/z
3. oh cheers!
Thing is I didn't know about enlarging/ shifting loci, the edexcel book doesn't seem to mention a lot about it. I shall ponder on your answer, see if I can get it all in time for Tuesday's exam!
4. (Original post by oceane)
oh cheers!
Thing is I didn't know about enlarging/ shifting loci, the edexcel book doesn't seem to mention a lot about it. I shall ponder on your answer, see if I can get it all in time for Tuesday's exam!
Lol! No the P6 book doesn't mention it as far as I 'm aware of. I'm doing P6 on tuesday as well and I've never seen it come up in past papers although saying that my teacher only had two that he could give me!! Oh well. Good luck on tuesday!!
5. Yeah, the Edexcel P6 book doesn't cover it at all!
I had to look at some AQA P6 papers/mark schemes to get a general feel for the topic
6. Ok then, it's over to the AQA past papers site. Think I'll have yet another cup of tea first though...

Good luck to you too guys!!
7. there was a question last summer (edexcel) question 6 (last part of it) about transformations with matrices... i dont understand it at all.

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