(Original post by insparato)
STEP III Question 6
A complex number z lying on a circle centre K and radius where K represents the complex number k.
-k*z - kk* = ax + aiy - bix - by - ax - aiy + bix - by = -2ax - 2by
The locus of P is which represents the complex number z
| z - i | = 1
let w_1 = x + iy
The locus of W_1 is a circle with centre (0,-i) and radius 1 on the argand diagram.
Thus the locus L is
|w_1 + i | = 1
Now this is where it gets tricky.
w_2 = z*
This formula i proved is screaming at me.
We know it has to be some circle and you can guess as its going to be of radius one if its the conjugate of z.
Im tempted to just say if locus of z is |z - i | = 1
therefore the locus of the conjugate of z is |z* - i | = 1
this incidentally happens to come out as the locus of L.
Just had another thought. What is the conjugate of a complex number?, as far as im aware its a reflection in the x axis of the original complex number on the argand diagram. So if the locus of Z is a circle, if you take all the specific points you could make its conjugates by reflecting in the x axis.
So if the locus of Z is a circle centered at (0,i) and has a radius of 1 it touches(0,0) a reflection of this is simply a circle centered (0,-i) with radius 1.
This happens to be x^2 + (y + 1)^2 = 1
Which is the same locus as L.
The next part is bugging me here,
Determine the positions of P for which Q1 and Q2 coincide.
I can see how the points Q1 and Q2 could coincide they are on the same locus. However the positions of p ? Am i suppose to find where Q1 and Q2 coincide and use one of the formulas given to find where it transforms onto the Z plane?
Ive proved a formula i have not yet used, some how i think this has got to come in here.