Given is one "random" point on the ellipse, the eccentricity and the center of the ellipse. How can I calculate the semi-major axis?
I'm not very math-savvy, so I hope the problem is easy to solve. Thanks a lot.
Edit: In some cases the ellipse is rotated on the y-axis.
Edit 2: So thanks to Cadenza I can calculate the semi-major axis if the ellipse is in a 2D space but what can I do if the ellipse is in a 3D space and rotated on the y-axis?
On a german Q&A website somebody said that it's impossible with these given informations. Is that true? If yes which informations except the semi-minor axis could be helpful?
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$\begingroup$Any "random" (a better word in this case is arbitrary) point $(x,y)$ on an ellipse whose axes are parallel to the $x$- and $y$-axes satisfy the equation
$$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$$
where $(h,k)$ denotes the center of an ellipse, and $a$ and $b$ denote the semi-major and semi-minor axes ($a$ is semi-major if the ellipse is horizontal).
The eccentricity of an ellipse the ratio of $c$ to the semi-major axis (so if the ellipse is horizontal, the eccentricity is $\frac{c}{a}$). Here, $c$ is the focus of the radius, given by $a^2 = b^2 + c^2$ (again, horizontal case).
You are given $(x,y)$, $(h,k)$, and $\frac{c}{a}$ (or $\frac{c}{b}$). Can you use these equations to recover $a$ (or $b$)?
Note that there is also the case in which the ellipse is rotated (with respect to the standard axes), in which case you would have to consider a more complicated equation (see: Rotation of axes).
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