Prove y = 0 or x^2+y^2=1 if z = x+iy is a non zero complex number and z+1/z=k;k is real

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NCEA Level 3 - Excellence question - Complex number - If z = x+iy and z+1/z=k (k is real), then prove that y = 0 or x^2+y^2=1
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Is it not simpler just to rationalize (1/(x+yi)) on its own and then end up with
k= x+yi+((x-yi)/(x^2+y^2)) therefore (x^2+y^2) must equal 1 in order for the +yi and -yi to cancel, or y=0 so that all i components are canceled?
Or is that incorrect?

oliviaflanagan