Loci in Complex Plane

walrein_lim88

antimatter

For the first part, i think z is a circle with radius 1. We know that for z = R cos theta + i R sin theta, z2 = R2 cos 2theta + i R2 sin 2 theta. Therefore, we know that point Q is actually a bigger circle which has a radius of 3 unit and has a phase (theta) of twice of that of P. We know that the locus of Q has the same center as the locus of P which is at the origin. The minimum is when P and Q are in the same phase, to get that, we know that theta must be zero. It should turn out that the distance is 2 unit. On the other hand, the distance between P and Q should be maximum when the phases of P and Q are seperated by pi. To get that, you actually need your theta to be pi. Hence, i think you will get that the maximum distance between P and Q is 4.

antimatter

For the second part, i think you just express z = Re + i Im, make the denominator real by multiplying both top and the bottom with the denometor's conjugate, and then take the i)length and ii)argument to compare with the right hand side (2 for part (i) and half pi for part (ii)). I think then you will get some function of Re in terms of Im. It will then be the locus you are looking for.

數學神童

walrein_lim88, these questions are from further math past year paper, is it?

walrein_lim88

回复 4# 數學神童


    是的。。。。

DADDY_MUMMY

神童，用华语。我英文不好。

多普勒效应

请用中文发表, 谢谢.