[Azland]

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

[ghostwalker]

(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.

[Scorpionfish]

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.

[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?

[Scorpionfish]

(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.

[Azland]

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

[ghostwalker]

(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.