Where z is a complex number, what is the cartesian form of |Z-2+3i| = 1?
Z is simply a general complex number, which can be written as Z = x+iyHere |Z-2+3i| = 1 can be written as |Z-(2-3i)| = 1, which is just an expression for every Z whose distance from the point (2,-3i) is equal to 1.We can solve this by recalling that Z = x+iy, and so we can seperate the real and imaginary parts in the modulus function. i.e. 1 = |(x-2) + i(y+3)| Evaluating the modulus now becomes simple as we calculate the magnitude using pythagoras. This Yields:12 = (x-2)2 + (y+3)2 , which is the cartesian form!we recognise this as the equation of a circle, with centre (2,-3) and radius 1.
