$\text{ Number of solutions of }|z-1|+|z+i|=4\text{ and }|2z-1+i|=\sqrt{14}\text{ is }$

$|z-1|+|z-i|=4\text{ represents ellipse with foci }(1,0)\text{ and }(0,-1)\text{ and length of major axis }=4$

$\begin{array}{l}\text{ Centre of ellipse is }\left(\frac{1}{2},-\frac{1}{2}\right)\text{ and }\left(\frac{1-i}{2}\right)\\ 2a=4,2ae=\sqrt{2}\Rightarrow e=\frac{1}{2\sqrt{2}}\\ \text{ From }{b}^2=a^2\left(1-e^2\right),\text{ we get }b=\sqrt{\frac{7}{2}}\end{array}$

$\begin{array}{l}\text{ Now }|2z-1+i|=\sqrt{14}\\ \Rightarrow \left|z-\frac{1}{2}+\frac{i}{2}\right|=\sqrt{\frac{7}{2}}\\ \Rightarrow \left|z-\left(\frac{1-i}{2}\right)\right|=\sqrt{\frac{7}{2}}\text{ represents a circle having centre at }\frac{1-i}{2}\text{ and radius }=\sqrt{\frac{7}{2}}\text{. }\end{array}$

Since radius of circle = minor axis of ellipse,
Hence number of solution of given equation= 2.