Given, $|z-3 i|+|z+5 i|=4$
It is of the form $\left|z-z_1\right|+\left|z-z_2\right|=k$ $\ldots$ (i)
Eq. (i) represents an ellipse if $k>\left|z_1-z_2\right|$
Here, $z_1=3 i$ and $z_2=-5 i$
$\begin{aligned} & \Rightarrow\left|z_1-z_2\right|=|3 i+5 i|=|8 i|=8 \text { and } k=4 \\ & \text { But } k < \left|z_1-z_2\right|\end{aligned}$
Hence, no such point $z$ exists.