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The complex equation $|z-8-2i|+|z-5-6i| = 5$ looks like it is an equation of an ellipse but in reality it represents a line segment, why?

The equation of an ellipse is $|z-z_1|+|z-z_2|=2a$, where $z_1$ & $z_2$ are the focus points and $2a$ is the length of the major axis. The distance between $z_1$ & $z_2$ is $5$ which is equal to the major.

Please help me to understand under what conditions the equation $|z-z_1|+|z-z_2|=2a$ represents an ellipse.

Gary
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Samar Imam Zaidi
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  • https://math.stackexchange.com/questions/1126121/defining-the-equation-of-an-ellipse-in-the-complex-plane – Marcos Mar 10 '22 at 08:49
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    if $|z_1-z_2|=2a$, like in your case here, then you have a degenerate situation. An infinitely thin ellipse, ie. a line segment. – Michal Adamaszek Mar 10 '22 at 08:51
  • The geometry of $\Bbb C$ is the Euclidean plane $E$. If $B,C$ are unequal points in $E$ then for any point $A$ we have $|AB|+|AC|=|BC|$ iff $A$ belongs to the line-segment $BC$. Here, $A=z$ and $B=8+2i$ and $C=5+6i$ and $|BC|=|(8+2i)-(5+6i)|=5.$ – DanielWainfleet Mar 10 '22 at 10:22

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