# Thread: complex number locus

1. ## complex number locus

i really need help on this...
if z is an imaginary number, and a is a real parameter, sketch the locus of;
z-5 = ai(z+5)
i know how it would work if it were moduli, but not a clue with just the imaginary numbers as they are
thanks

2. Hello,
Originally Posted by allazkatraz
i really need help on this...
if z is an imaginary number, and a is a real parameter, sketch the locus of;
z-5 = ai(z+5)
i know how it would work if it were moduli, but not a clue with just the imaginary numbers as they are
thanks
Write $z=x+iy$
And substitute in the equation.

x and y will be the coordinates of z in the complex plane, so an equation between the two will give you the locus

3. ## re:

ok, i understand that part, but then i get to this point;

y= [10+x(ai-1)] / (a+i)

and i don't know what to do with it..
i know that the answer is supposed to be a circle, but how do i get rid of the a's in my equation? do i even need to?
thanks for the help so far

4. Originally Posted by allazkatraz
ok, i understand that part, but then i get to this point;

y= [10+x(ai-1)] / (a+i)

and i don't know what to do with it..
i know that the answer is supposed to be a circle, but how do i get rid of the a's in my equation? do i even need to?
thanks for the help so far
If you do what moo suggests you will get

x + iy = -ay + i(ax + 5a).

Equate real and imaginary parts on each side:

x = -ay => a = -x/y .... (1)

y = ax + 5a = a(x + 5) .... (2)

Substitute (1) into (2) etc.