Real and imaginary cases with a conjugate

The complex number $w$ is such that $kw^2+kww^*+iw-iw^*-1=0$, where $w^*$ is the complex conjugate of $w$ and $k$ is a real and non-zero constant.

(i) For $w=a+bi$ where $a$ and $b$ are real numbers, obtain an expression for $b$ in terms of $a$ and $k$. Explain why $w$ is either purely real or purely imaginary.

(ii) Using your result in part (i), or otherwise, find the real roots of the equation $2w^2+2ww^*+iw-iw^*-1=0$.