locus question (1 Viewer)

[theMoment]

The complex number z moves such that Im[1/(zbar -i)] = 2.

Show the locus is a circle and find its centre and radius.


Help please?

 

[MOP777]

That should be all correct mate, when you 'realise' the denominator, if its not clear, you multiply the top and the bottom of the fraction by the conjugate of the denominator, ie multiply the top and bottom of 1/a-ib by a+ib and it gives you the fraction as shown above.

 

[theMoment]

I don't understand the Im bit.

When you get to the Im, you let top x = 0 and the i is removed.
I don't get that.

 

[MOP777]

I don't understand the Im bit.

When you get to the Im, you let top x = 0 and the i is removed.
I don't get that.

Oh k sorry about that.

So when z = x+iy

(x and y are real numbers)
and i = sqrt(-1)

z has 2 componants to it, the real part and the imaginary part.

so Re(z) = x
Because Re(z) means the real part of the complex number z.

The imaginary part is the part in front of the i. So Im(z) is the Imaginary part of the complex number z and Im(z) = y
Because y is the part with i's in it.

So if z = x+iy
and the Im(z) = 2

Then z = x +2i

Or in another case if Im(z) = 200+5y
and if Re(z) = 150+2x-2y

then z = Re(z) + Im(z)*i

so z = 150+2x+2y + i(200+5y)

Hope that clears things up.

Do you understand it now?