*Diagonal* section in a cone (the section should be an *Ellipse*); but actually a *Circle* how come?!

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In the picture you see an oblique cone with two sections through it ($AB$ and $CD$ segments).

I computed the semi-major and semi-minor axes and both of them were $\dfrac{rh^3}{l^3}$ in the $AB$ case. so this is in fact a circle?

Also, in the $CD$ case, the semi-major axis is longer than $AB$ case; but its semi-minor axis is obviously $\dfrac{rh^3}{l^3}$ again. so it is an ellipse.enter image description here

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When you have an oblique circular cone like this one (and it is an oblique circular cone, since the axis--then line from the vertex to the center of the circular base--is not perpendicular to the base), all planes parallel to the base (such as the ones that cut the cone along $AD$ and $BC$) intersect the cone in circles; but there is also another family of planes whose intersections with the cone also are circles.

You have found one of that other set of planes for your particular cone.