Solution
= Solution
Write $z=x+iy$ and use the <complex conjugate> $z^*=x-iy$. Then
$$
u=\frac32(x+iy)+\frac12(x-iy)=2x+iy.
$$
Thus the <real part> is stretched by a factor of two while the <imaginary part> is unchanged. If $u=X+iY$, substitution into the <circle> equation gives
$$
\boxed{\frac{(X+2)^2}{16^2}+\frac{(Y+1)^2}{8^2}=1.}
$$
This is an <ellipse> with centre $(-2,-1)$, horizontal semiaxis $16$ and vertical semiaxis $8$. It is the second panel in the figure in part (b).