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Is it possible to determine a Conic given two points on the conic and equation of major and minor axis?

I choose $5$ random points on $\mathbb R^2$ independently. Since 5 points determine a conic, I get hold of a circle, parabola, ellipse or a hyperbola. Of course, it is most likely a hyperbola or an ellipse, probability of a parabola or circle is almost 0.

Now given two of these 5 points and equations of major, minor axis can I reach back to the ellipse/hyperbola?

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It depends.

Take an example.

Whatever major and minor axis are given to you, shift and rotate axes such that the major axis is new x-axis and minor axis is new y-axis. Now, if you are given points, say, $(3,0)$ and $(-3,0)$ (in new coordinate system). It may be ellipse with equation $$\frac{x^2}{9}+\frac{y^2}{k^2}=1$$ (for some k) or it may be a hyperbola with equation $$\frac{x^2}{9}-\frac{y^2}{k^2}=1$$

But, say, if your points are like $(3,0)$ and $(0,2)$, you know it is ellipse with equation, $$\frac{x^2}{9}+\frac{y^2}{4}=1$$

Hope it helps:)

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You can reflect the two points in the major or minor axes (or in both)

  • Usually this will lead to eight points, which is more than enough to find the conic (assuming it exists)

But sometimes these points are not distinct, and you end up with two, four or six points

  • Six points is enough (two will lie on an axis)

  • Four points may be enough if they all lie on the axes, but not be if they do not

  • Two points (on an axis) will not be enough

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