The question goes like this: On an Argand diagram, sketch the locus representing complex numbers $z$ satisfying $|z+i|=1$ and the locus representing the complex numbers $w$ satisfying $\arg(w-2)=\dfrac{3π}{4}$. Find the least value of $|z-w|$ for points on these loci.
I know how to do the first two sketches; a radius 1 circle in coordinate $(0,-1)$ and a line starting from $(2,0)$ at that angle from the origin, but how do you find the last one? The answer has an exact value, so I can't do it by looking at the diagram. Thanks in advance.
Best Answer
Well... you have that $x-2 = -y$, where $y>0$. This is just a line. Well, $w \equiv (-v+2) + vi$ and $z = \cos(\theta) + (\sin(\theta)-1)i$. So, $|z-w|^2 = (-v+2-\cos(\theta))^2 + (v+1-\sin(\theta))^2$, so we can easily optimise by choosing $v-\sin(\theta) = 1$ and $v+\cos(\theta) = 2$ yields $\cos(\theta) + \sin(\theta) = 1$ and so $\sin(\theta+\pi/4) = 1/\sqrt{2}$ and so $\theta = 0$ or $\theta = \pi/2$. This then implies that $v = 1$ or $v = 2$ respectively. Now fill in the details.