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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.

Max Wong
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George
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3 Answers3

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Their loci in the $xy$-plane are, respectively,

$$x^2+(y+1)^2 = 1, \>\>\>\>\> x+y-2=0$$

The least value of $|z-w|$ for the points on the two foci is equivalent to the shortest distance between the circle and the line. Since the distance from the center of the circle $(0,-1)$ to the line $x+y-2=0$ is

$$d=\frac{|0\times1+(-1)\times 1 -2|}{\sqrt{1^2+1^2}} = \frac{3}{\sqrt{2}}$$

The least value is just

$$ d-r = \frac{3}{\sqrt{2}} -1 $$

Quanto
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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.

Christopher K
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The general situation here is that you have a variable point on a circle ($z$) and a variable point on a line ($w$). What you're being asked to do is minimize $|z-w|$ or bring the points as close as possible. Now ask yourself the question, how close can a circle and a line get. If the line is a tangent or a secant to the circle, the answer is $0$ since they are practically touching each other. However, if the line does not touch the circle (as in this case), the closest they can get if the perpendicular distance of the centre of the circle to the line minus the radius of the circle. Hopefully you can work out what you need to do now from this clue.

sayantankhan
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