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# FM Complex Number Question Help!

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1. FM Complex Number Question Help!
New Question : Help please.

The variable complex number z is given by
z = 2cosθ + i(1 − 2 sinθ) for −pie < θ < pie

Show that |z − i| = 2

Prove that the real part of

1 / z + 2 − i is constant for −pie < θ < pie
Last edited by Azland; 02-04-2012 at 19:31.
2. Re: FM Complex Number Question Help!
(Original post by Azland)
The complex number u = -1 - i

Sketch an Argand diagram showing the points representing the complex numbers u and u^2.

Shade the region whose points represent the complex numbers z which satisfy both the inequalities |z| < 2
and |z − u^2 | < | z - u |

Just lost on what to do in this question regarding the inequality.
For your 2nd inequality.

It's perhaps easier to first consider |z-u^2|=|z-u|, i.e. the distance of z from u^2 is equal to the distance from you. This will be the perpendicular bisector between the two points representing u and U^2.

Since you actually want the distance to u^2 to be less, then the points satisfing this inequality will be the half plane closer to u^2.

Your first inequality is just points inside the circle radius 2, centred at the origin.

Then shade the overlap of the two.
3. Re: FM Complex Number Question Help!
The first inequality is all complex numbers whose modulus is less than 2, i.e. all complex numbers that are within 2 of the origin on an argand diagram. So, this will be a circle (shaded in) about the origin, with radius 2.

The second inequality is saying that the distance in between z and u^2 is less than the distance between z and u - i.e. all points closer to u^2 than u. So, if you imagine a perpendicular bisector in between u and u^2, and then shade everything on the side that is closer to u^2.

Points that satisfy both inequalities will be the overlap between the two shaded regions - i.e. the part of the circle that is on the u^2 side of the perpendicular bisector.

Hope this helps.
4. Re: FM Complex Number Question Help!
Ah thanks got it. I was trying to solve this using algebra but that didnt seem to work out and to be honest takes so long. whenever a question says z - u, I can directly just plot the points for u and work out the inequality then?
5. Re: FM Complex Number Question Help!
(Original post by Azland)
Ah thanks got it. I was trying to solve this using algebra but that didnt seem to work out and to be honest takes so long. whenever a question says z - u, I can directly just plot the points for u and work out the inequality then?
Yeah, and I would say normally when it starts talking about shading regions of an argand diagram, it's about looking at the inequalities and recognising the form, rather than doing fiddly algebra.
6. Re: FM Complex Number Question Help!
New Question : Help please.

The variable complex number z is given by
z = 2cosθ + i(1 − 2 sinθ) for −pie < θ < pie

Show that |z − i| = 2

Prove that the real part of

1 / z + 2 − i is constant for −pie < θ < pie
7. Re: FM Complex Number Question Help!
(Original post by Azland)
The variable complex number z is given by
z = 2cosθ + i(1 − 2 sinθ) for −pie < θ < pie

Show that |z − i| = 2
I'd start by working out what z-i is.

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Last updated: April 2, 2012
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