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Conic sections are formed by the intersection of a cone an plane. But how do we know that the shape developed by the plane is the same which we define in other ways.what i mean is that suppose the case of an ellipse . Let the plane intersection the cone at such an angle that an ellipse is formed.

Now prove to me that this shape is actually an ellipse by the definition that sum of distances from 2 points is constant. Similiarly prove for other definitions.

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    $\begingroup$ You want to take a look at Dandelin spheres. $\endgroup$ – Blue Feb 15 at 2:11
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    $\begingroup$ Search for the ice cream cone proof for the ellipse liorpachter.wordpress.com/2016/05/09/the-ice-cream-cone-proof $\endgroup$ – Ethan Bolker Feb 15 at 2:12
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    $\begingroup$ There's an explanation of this with Dandelin spheres here too: math.stackexchange.com/a/137462 $\endgroup$ – David K Feb 15 at 6:11
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    $\begingroup$ For a hyperbola, one Dandelin sphere is positioned similarly to the one for the ellipse that's closer to the vertex. The second sphere is found by extending the cone through the vertex to make it a double cone and fitting the sphere into the extended part of the cone. The fact that the vertex now falls between the two spheres is what geometrically converts the sum to a difference. $\endgroup$ – Oscar Lanzi Feb 15 at 10:33
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    $\begingroup$ For a parabola: There is only one Dandelin sphere. It is tangent to the cutting plane at the focus. It is also tangent to the cone on a small circle whose plane intersects the cutting plane at a line. This line is the directrix. $\endgroup$ – Oscar Lanzi Feb 15 at 10:38