if x + iy = 1/x - iy, what is always true about x and y?

Question

a. x^2 + y^2 = 0
b. x^2 + y^2 = 1
c. x^2 - y^2 = 0
d.x^2 + y^2 = 1
e. xy = 0

Answer 1

(x + i y) (x - i y) = 1
x ² - (i ²) y ² = 1
x ² + y ² = 1
OPTION d

Answer 2

equate the real part of LHS to real of RHS
x=1/x
so, x^2=1
equate the imaginary part of LHs to imaginary of RHS
y=-y
so, y+y=0
2y=0
so, y=0
substitute x^2=1 & y=0 in x^2 + y^2
=1+0^2=1
so, b& d are true
also x*y=sqrt(1) * 0 = 0
so, b,d,e are true

Answer 3

a , i think not sure...