maths complex numbers

Question

Given that the real and imaginary parts of the complex number z = x + iy satisfy the equation (1 + 2i)x + (2 − 3i)y = 3 − i.

(i) Find x and y.

(ii) Express z in polar form, and hence find z 10 .

Answer 1

Hi Tina,

your equation (1 + 2i)x + (2 − 3i)y = 3 −i is actually two equations in one, one each for real and imaginary parts of z:

x +2y = 3 and (2x -3y)i=-i

(i) solving these two equations x+2y=3 and 2x-3y=-1 using the usual methods you will find x=1 and y=1 hence z = 1+i

(ii) polar representation of a complex number z=x+iy is z=r cos (theta) +I sin (theta). Here, r=sqrt(1^2+1^2)=sqrt(2) and theta=Pi/4=45 deg angle. Hope that helps.