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# Complex Numbers - mapping watch

1. This topic is fairly new to me and I don't know how to go about doing this question:

Show that the mapping

where and is a real number, maps the circle in the plane into an ellipse in the plane and find its equation.

I have found and in terms of and but don't know how to progress from here.
2. Go the other way Find x and y in terms of u and v. Then substitute into |z| =1, i.e. x^2 +y^2 =1 to get a quadratic equation in u and v; hopefully an ellipse.
Chris
3. Do you have
and ?

Rearrange these to find x and y in terms of u, v and c then use the fact that Which is the locus of z.
4. What did you get for u and w? (I would start by assuming |z| = 1 and so writing z = cos t + i sin t).
5. (Original post by DFranklin)
What did you get for u and w? (I would start by assuming |z| = 1 and so writing z = cos t + i sin t).
I got

Which looked nasty and then I got scared and came onto TSR

I though about trying to find u and v in terms of x and y but that seems a little complex (sorry for the awful pun)

I imagine that the z = cost + i sint is the way to go - I'll give it a whirl and get back to you.
6. I don't think your functions for u and v are right. Want to post your working?
7. (Original post by The Muon)
I got

Which looked nasty and then I got scared and came onto TSR

I though about trying to find u and v in terms of x and y but that seems a little complex (sorry for the awful pun)

I imagine that the z = cost + i sint is the way to go - I'll give it a whirl and get back to you.
You should write z = x + iy straight away and use the conjugate to get rid of anything complex in the denominator and you should be able to get w in the form a + bi with a and b some function of x and y. And you can then use the condition abs(z) = 1.

On an aside the question should have stipulated c is not zero.
8. (Original post by DFranklin)
I don't think your functions for u and v are right. Want to post your working?
I'd rather not as it is a huge chunk that looks like it will lead no where but following on from the cost+isint approach.

I found

Would this be the equation of the new ellips in parametric form? So what I would have to do is just convert to Cartesian?
9. (Original post by The Muon)
I got

Which looked nasty and then I got scared and came onto TSR

I though about trying to find u and v in terms of x and y but that seems a little complex (sorry for the awful pun)

I imagine that the z = cost + i sint is the way to go - I'll give it a whirl and get back to you.
I'm pretty sure you should have and
Since , both expressions can be simplified. From there rearrange for x and y and substitute into

Spoiler:
Show
I believe
10. (Original post by Mathletics)
I'm pretty sure you should have and
Since , both expressions can be simplified. From there rearrange for x and y and substitute into

Spoiler:
Show
I believe
Yeah, I made a slip in my calculation and forgot to square my x and y when simplifying my denominator.

I have now got the right answer (albeit by Dave's method)

Thanks guys - rep will be handed out over the next few days

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