| z - 12j | = 3 | z + 36 | if z is written in the form z = x+jy. What is the equation of the locus for z as a cartesian equation in terms of x and y
Cheeeeers
Geometrically, |z- a| is the distance from point z, in the complex plane, to point a. So |z- i| is the distance from the generic point z to i while |z+ 36| is the distance from z to -36. The locus is the set of points whose distance from i, (0, 1) in "xy" terms, is three times its distance from -36, (-36, 0) in "xy" terms.
The distance from (x,y) to (0, 1) is $\displaystyle \sqrt{x^2+ (y- 1)^2}$ and the distance from (x, y) to (-36, 0) is $\displaystyle \sqrt{(x+ 36)^2+ y^2}$ so "|z- i|= 3|z+ 36|" is equivalent to $\displaystyle \sqrt{x^2+ (y+1)^2}= 3\sqrt{(x+ 36)^2+ y^2}$. Square both sides and the simplify.