On an argand diagram, sketch the locus representing complex numbers $z$ satisfying $|z+i|=1$ and the locus representing complex numbers $w$ satisfying arg$(w-2)=\frac{3}{4}\pi$ find the least value of $|z-w|$ for points on these loci.
I have sketched the loci. And can anyone teach me how to find the least value with a diagram? Thanks
Best Answer
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See the above diagram. Here, note that $\triangle{ABC},\triangle{AOD}$ and $\triangle{DEF}$ are triangles with $45^\circ,45^\circ,90^\circ$ and with $|AC|=|AO|=|OD|=1,|ED|=|EO|-|OD|=2-1=1$.
Thus, $$\begin{align}\text{the least value of $|z-w|$}&=|CF|\\&=|CD|+|DF|\\&=|AD|-|AC|+\frac{1}{\sqrt 2}|DE|\\&=\color{red}{\frac{3}{2}\sqrt 2-1}\end{align}$$