Question

1. z is a variable complex number such that mod(z) = 2. Show that the point z + (1/z) lies on an ellipse of eccentricity 4/5 in the complex plane.

Answer 1

Let z=x+iy where x, y are real variables. |z|=√(x^2+y^2)=2; x^2+y^2=4 (equation of circle radius 2 with centre (0,0)).

1/z=1/(x+iy)=(x-iy)/(x^2+y^2)=(x-iy)/4=x/4-iy/4.

Let Z=z+1/z=(5/4)x+(3/4)iy=(1/4)(5x+3y). |Z|=(1/4)√(25x^2+9y^2) which indicates the relationship between x and y for complex Z.

x^2/A^2+y^2/B^2=1 is an ellipse, where A is the length of the semimajor axis and B is the length of the semiminor axis, assuming A>B. The eccentricity, e=√(A^2-B^2)/A, so e^2A^2=A^2-B^2 or B^2=A^2(1-e^2). If B>A then A and B need to be interchanged.

Z^2=(25x^2+9y^2)/16; since 16/9>16/25, B is the semimajor axis and A the semiminor axis, so e=√(16/9-16/25)/(4/3)=(3/4)√(16(16)/225)=(3/4)(16/15)=(4/5).