Solving Locus Equation: |z-12i|/|z+36|=3

[captainquarks]

Hey guys, this is my problem:

find the cartesian equation of the locus:

|z-12i|/|z+36|=3

So far I've got here:

introduce z=x+iy then collect terms gives:

√(x²+(y-12)²)=3√((x+36)²+y²)

Squaring both sides I get:

x²+(y-12)²=9(x+36)²+9y²

This is where I am stumped... I've tried expanding it to see if I can factorise it into a standard circle, but I havn't been able to find a neat solution. Am I missing a trick here? Any help would be hugely appreciated as It's actually driving me insane!

Thanks

 

[tiny-tim]

hey captainquarks!

hint: complete the square

 

[captainquarks]

Hey, thanks tiny-tim for your time and reply :), since I posted this I've had a good attempt at a solution... Let me know what you think perhaps?

(x+40.5)^2 + (y+1.5)^2 = (405/2)^0.5

I did this, as you said, by completing the square:

i worked with this equation: x^2 + 81x + y^2 + 3y + 1440 = 0

Any input is vastly appreciated, thank you :)

 

[tiny-tim]

hey captainquarks!

i'm just going to bed :zzz:, and i haven't had time to check your figures

have you noticed that the original question is another way of writing "the distance from point a is 3 times the distance from point b" ?

goodnight! ​