# Thread: Help needed with complex numbers!!

1. ## Help needed with complex numbers!!

I am stuck on this real complex question- i have done the first 3 parts but m stuck on the last- can't figure out the right shaded portion.

The complex number -2+i is denoted by u.
a) Given that u is a root of the equation x^3-11x-k=0, where k is real, find the value of k.
b) Write down the other complex root of this equation.
c) Find the modulus and argument of u.
d) Sketch an Argand diagram showing the point representing u. Shade the region whose points represent the complex numbers z satisfying both the inequalities
|z|<|z-2| and 0<arg(z-u)<π/4

So far i have reached 3 inequalities
1) y<x+3, 2)y<1 and 3) x<1

Can somebody please help me with this question. If i am doing anything wrong plz correct me.

2. ## Re: Help needed with complex numbers!!

Originally Posted by IceDancer91

The complex number -2+i is denoted by u.
a) Given that u is a root of the equation x^3-11x-k=0, where k is real, find the value of k.
b) Write down the other complex root of this equation.
c) Find the modulus and argument of u.
d) Sketch an Argand diagram showing the point representing u. Shade the region whose points represent the complex numbers z satisfying both the inequalities
|z|<|z-2| and 0<arg(z-u)<π/4

So far i have reached 3 inequalities
1) y<x+3, 2)y<1 and 3) x<1
Here is a start: $(-2+i)^3-11(-2+i)=20~.$
Now you show of your own work!

3. ## Re: Help needed with complex numbers!!

Oh sorry i forgot to tell that i have solved the first 3 parts and i m only not able to solve the last part of this question... :/

4. ## Re: Help needed with complex numbers!!

|z| is the distance from z to 0. |z- 2| is the distance from z to 2. The set of all point such that |z|= |z- 2| is the perpendicular bisector of the interval from 0 to 2. The set of points, z, such that |z|< |z- 2| is the set of all points on the same side of that bisector as 0.

Arg(z) is the angle the line from 0 to z makes with the real axis. Arg(z- u) is the angle between the line from 0 to u and the line from 0 to z. The set of all points, z, such that $0< Arg(z-u)< \frac{\pi}{4}$ is the area bounded by the line through u and the line at 45 degrees to that line.

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### shade the region qhose points represent complex numbers z which satisfy both the inequalities|z-2i|≤2

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