 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2016-resit/1/ii/solution

Put

z = x + i y

and

z^{*} = x - i y

. The first two squared terms are complex conjugates, so their sum is

2 [(x - 2)^{2} - (y + 1)^{2}]

. Also

z z^{*} = x^{2} + y^{2}

. The equation therefore reduces to

2 [(x - 2)^{2} - (y + 1)^{2}] + 2 (x^{2} + y^{2}) + 6 = 4 (x^{2} - 2 x - y + 3) = 0.

Thus the locus is the upward-opening parabola

y = (x - 1)^{2} + 2,

1 + 2 i

and axis

x = 1

 New to topics? Read the docs here!