SOLUTION: This is another question from a past Algebra & Complex Numbers paper which I am practicing for study for my Calculus exam. The question is apparently worth an excellence mark: Q

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Question 167782: This is another question from a past Algebra & Complex Numbers paper which I am practicing for study for my Calculus exam. The question is apparently worth an excellence mark:
QUESTION EIGHT
Find the equation of the locus of the point representing z if |z-3i|-|z+3i|=2.
I can work it out as far as:
|z-3i|-|z+3i|=2
(z=x+iy) => |x+i(y-3)|-|x+i(y+3)|=2
(|z|=+sqrt%28+x%5E2%2By%5E2+%29+) => +sqrt%28+x%5E2%2B%28y-3%29%5E2+%29+-+sqrt%28+x%5E2%2B%28y%2B3%29%5E2+%29+=2
+sqrt%28+x%5E2%2By%5E2-6y%2B9+%29+-+sqrt%28+x%5E2%2By%5E2%2B6y%2B9+%29+=2
but after that I get stuck.
Help appreciated.
Answer by oscargut(2103) About Me  (Show Source):
You can put this solution on YOUR website!
Let take w+=%28x%5E2%2By%5E2-6y%2B9%29
Let take z+=%28x%5E2%2By%5E2%2B6y%2B9%29
Note that w-z = -12y
then
sqrt%28w%29-sqrt%28z%29=2
then
%28sqrt%28w%29-sqrt%28z%29%29%5E2=2%5E2
w%2Bz-2sqrt%28w%29sqrt%28z%29=4
then
w%2Bz-4=2sqrt%28w%29sqrt%28z%29
then
%28w%2Bz-4%29%5E2=%282sqrt%28w%29sqrt%28z%29%29%5E2
then
w%5E2%2Bz%5E2%2B16%2B2wz-8w-8z=4wz
w%5E2%2Bz%5E2%2B16-2wz-8w-8z=0
%28w-z%2B4%29%28-z%2Bw%2B4%29-16w=0
then
%28-12y%2B4%29%2812y%2B4%29=16%28x%5E2%2By%5E2-6y%2B9%29

144y%5E2%2B16=16%28x%5E2%2By%5E2-6y%2B9%29

16x%5E2-128y%5E2-96y%2B128=0