Drawing a Cone on a Plane If we draw a cone on a plane, the picture usually consists of an ellipse, two line segments sharing one common endpoint, while the other end point connected to each vertex of major semi-axes of the ellipse. The problem is that, the two line segments cannot be tangent to the ellipse at the vertexes since otherwise they will be parallel and do not intersect at a common point. Somehow one can sense a "problem" here: Each line segment needs to intersect the ellipse one more time on the upper half but this is generally never seen in pictures of cones. Can anyone give an explanation?

 A: Forgive my grammar and logic 
I'm assuming that It's really an ellipse.
That's because point where tangent touches the ellipse and tangent at diametrically opposite points are close enough to fool our eyes, and if you increase height of the cone you'll see them being indistinguishable and when $\text{height } \to \infty$ Those lines will coincide.
I hope this Will help

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A: When a curved surface $S$ is projected onto a plane there will be in general points $p\in S$ for which the projection ray through $p$ is lying in the tangential plane $T_p$. These points are singular points of the projection map $\pi:\>S\to{\mathbb R}^2$, because $\pi$  is not locally bijective near $p$, and their images $\pi(p)$ form the visible outline in the two-dimensional picture. When $S$ is a sphere  then the outline is a circle under orthogonal parallel projection, and an ellipse in other cases.
In the case of an (infinite) cone $C$ the outline is in general a pair of lines intersecting at $\pi(s)$, where $s$ is the tip of $C$. 
The original ellipse $\epsilon$ in your figure is a curve (a circle) on $C$ which intersects the preimage of the outline (the set of singular points of $\pi$) in two  points $p$ and $q$. It is a theorem that (except in very special cases) the image curve $\pi(\epsilon)$ will be tangent to the outline at $\pi(p)$, resp. $\pi(q)$.
A: The explanation is that your method of constructing a perspective projection of a cone is wrong: The line segments must not be connected to the axis of the ellipse (*); they must be tangents.
(*) except for the degenerate case where the projection is a triangle
