# Tagged Questions

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### Omitting $i$ in calculations

Is it possible in various calculations related to the complex plane which also include analytic geometry , calculating distances etc, to omit $i$ and treat the imaginary axis as simply the cartesian ...
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### Conic Sections and Complex numbers

If $\omega$ is a complex number such that |$\omega$| does not equal 1, then the complex number $$z = \omega + \frac{1}{\omega}$$ describes a conic. The distance between the foci of the conic described ...
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### Analytical Geometry problem with complex numbers - alternate solutions.

The question is to show that the equation of the lines making angles $45^\circ$ with the line: $$\bar{a}z + a\bar{z} + b = 0; \;\;\;\;\; a,z \in \mathbb{C}, b \in \mathbb{R}$$ and passing through a ...
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### Distance between point and line in the complex plane

Let $a,b$ be fixed complex numbers and let $L$ be the line $$L=\{a+bt:t\in\Bbb R\}.$$ Let $w\in\Bbb C\setminus L$. Let's calculate $$d(w,L)=\inf\{|w-z|:z\in L\}=\inf_{t\in\Bbb R}|w-(a+bt)|.$$ The ...
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### If $\left |z-3\right |=\left |z+i\right |$, where $z=x+iy$, prove that $3x+y=4$

If $\left |z-3 \right |=\left |z+i\right |$, where $z=x+iy$, prove that $3x+y=4$. I have got to the point where I have $\left |z \right |= \sqrt{x^2+(y+1)^2} = \sqrt{(x-3)^2+y^2}$ But really ...
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### Complex Numbers - Locus

Suppose that $k|z-z_1|=l|z-z_2|$ where $k\neq l$ and both are positive real numbers. Show that the locus of $z$ in the Argand diagram is a circle with center: $$\frac{k^2 z_1-l^2 z_2}{k^2-l^2}$$ and ...
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### Curve of Equal SWR

I'm trying to figure out how radio frequency "matching stubs" work. In order to fully understand the problem, I need to know how the "curve of equal SWR" looks like. I did a few plots, and it looks ...
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### Illustrations of a line and a curve intersecting for complex field

Are there nice illustrations on the Net of say $y=a·x+b$ and $y=x^2$ intersecting where x and y are complex? I'm thinking of the amplitude of y being depicted as height above the complex plane with ...
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### Locus of Z as cartesian equation

Could you please help with this locus problem? I think I am aiming for a cartesian equation in terms of $x$ and $y$ that may look like a circle equation e.g. $(x+a)^2 + (y+b)^2$ but I'm not sure. ...
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### Why do all circles passing through $a$ and $1/\bar{a}$ meet $|z|=1$ are right angles?

In the complex plane, I write the equation for a circle centered at $z$ by $|z-x|=r$, so $(z-x)(\bar{z}-\bar{x})=r^2$. I suppose that both $a$ and $1/\bar{a}$ lie on this circle, so I get the equation ...
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### How does this equality on vertices in the complex plane imply they are vertices of an equilateral triangle?

I've read that if the complex numbers $a_1$, $a_2$ and $a_3$ are the vertices of a triangle in the complex plane such that $$a_1^2+a_2^2+a_3^2=a_1a_2+a_2a_3+a_1a_3$$ then the vertices are actually ...
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### Given the cartesian coordinates of four points, how to calculate the interection of two lines they form?

Given four complex numbers $A, B, C, D$ interpreted as points on the plane, how can I calculate the number that represents the intersection of the lines formed by $A, B$ and $C, D$?
Let $r$, $s$ be positive real numbers and $\theta$, $\phi$ real numbers with $|\theta -\phi|<\pi$. Then an argument of $re^{i\theta}+se^{i\phi}$ lies between $\theta$ and $\phi$. Can someone give ...