It sounds like [tex]x,y[/tex] are supposed to be real numbers. If so, then we can do the following.
[tex]\dfrac1{x^2+y^2}+\dfrac1{x+yi}=1[/tex]
Multiply the second term's numerator and denominator by the conjugate of the denominator:
[tex]\dfrac1{x^2+y^2}+\dfrac{x-yi}{(x+yi)(x-yi)}=1[/tex]
[tex]\dfrac1{x^2+y^2}+\dfrac{x-yi}{x^2+y^2}=1[/tex]
[tex]\dfrac{x+1-yi}{x^2+y^2}=1[/tex]
Since the left hand side is equal to 1, this means it has no imaginary part, so that [tex]y=0[/tex]. Then the real parts of both sides of the equation give us
[tex]\dfrac{x+1}{x^2}=1\implies x+1=x^2\implies x^2-x-1=0\implies x=\dfrac{1\pm\sqrt5}2[/tex]
