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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

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  • $\begingroup$ Can you describe the equations of the loci? $\endgroup$ – thanasissdr Apr 2 '16 at 8:06
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$\qquad\qquad\qquad$enter image description here

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, $$\begin{align}\text{the least value of $|z-w|$}&=|CF|\\&=|CD|+|DF|\\&=|AD|-|AC|+\frac{1}{\sqrt 2}|DE|\\&=\color{red}{\frac{3}{2}\sqrt 2-1}\end{align}$$

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  • $\begingroup$ @Mathxx : The least value of $|z−w|$ is the shortest distance between a point on the circle represented by $z$ and a point on the line represented by $w$. So, we consider a line perpendicular to the line and passing through the center of the circle $A$. I hope that this makes it easier to understand the answer. $\endgroup$ – mathlove Apr 3 '16 at 7:15
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The shortest distance between two curves is along a common normal. Since one of the curves here is a line, it is easy to give an equation of its normal at each point. Then we just have to see which one is a normal to the other curve (a circle) as well. The minimum distance should be along this line.

Here is a diagram showing what's happening.

enter image description here

The solid line from the circle and the line is the one with the least distance. It's also a common normal. The dotted lines show what happens to the distance if you perturb this line slightly.

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