3. Let the complex number $z=x+y \mathrm{i}$ have real and imaginary parts $x, y$ that form the point $(x, y)$ on the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{16}=1$. If $\frac{z-1-\mathrm{i}}{z-\mathrm{i}}$ is a real number, then the complex number $z=$ $\qquad$.
Given 
$z=3 \cos \theta+4 \mathrm{i} \sin \theta$, then $\frac{z-1-\mathrm{i}}{z-\mathrm{i}}=1-\frac{1}{z-\mathrm{i}}=1-\frac{1}{3 \cos \theta+(4 \sin \theta-1) \mathrm{i}}$ $=1-\frac{3 \cos \theta-(4 \sin \theta-1) \mathrm{i}}{9 \cos ^{2} \theta+(4 \sin \theta-1)^{2}} \in \mathbf{R} \Rightarrow \sin \theta=\frac{1}{4}$, so $z= \pm \frac{3 \sqrt{15}}{4}+\mathrm{i}$.