Place a small sphere inside the cone, just below the plane. In fact the plane and the sphere are tangent at the point q.
Next, place another sphere in the cone, above the plane. This sphere is also tangent to the plane at q. Hence both spheres touch at the point q. These spheres resemble blobs of ice cream, scooped into an up-turned cone; hence this is called the ice cream cone proof.
The lower sphere intersects the cone in a lower ring, and the upper sphere intersects the cone in an upper ring. Let p be any point on the circle, where the cone and the plane intersect. Draw a segment from p down to the lower ring, and another segment up to the upper ring. These segments add up to a fixed distance d, the distance between the parallel rings. The segments are also equal to the distance from p to q. Thus the distance from p to q is half of d, all the way around the circle.
This isn't very interesting, until we tilt the plane. Tip the plane at an angle, so that it cuts the upper cone in an ellipse. Embed a small sphere below the plane, and a larger sphere above the plane. The spheres are tangent to the plane at the points q1 and q2. These become the foci for the ellipse.
Let p be any point on the ellipse. The distance from p to the lower ring, plust the distance from p to the upper ring, is a fixed distance d. Now the distance from p to the lower ring is the same as the distance from p to q1, and the distance from p to the upper ring equals the distance from p to q2. (If this is less than obvious, I'll prove it in a minute.) Therefore the distance from p to the first focus, plus the distance from p to the second focus, is constant. This defines an ellipse.
Ok, why is the distance from p to the upper ring equal to the distance from p to q2? In each case p is the start of a ray that runs tangent to the upper sphere. In one case the ray runs straight up the cone, and in the other the ray runs along the intersecting plane and just touches the sphere at q2. Let's move away from the ice cream cone and prove equality in its own setting. Let p be the head of two rays that run tangend to a common sphere. The points of tangency are x and y. The three points p x y determine a plane. Restrict attention to this plane, hence our sphere becomes a circle. This need not be the equator of the sphere; it could be a tiny circle near the north pole. No matter. The segments px and py are now tangent to the circle. Let c be the center of the circle and draw radii from c to x and y. Then draw the segment from c to p. This produces two right triangles with a common hypotenuse. Since the two legs formed by the radii are equal, the triangles are congruent, and px = py. That completes the ice cream cone proof.
The same proof works for a cylinder. Let a plane cut across a vertical cylinder at an angle, intersecting in an ellipse. Place two spheres inside the cylinder, above and below the plane. The points of tangency become the two foci. The spheres intersect the cylinder in an upper and lower ring. From here the proof is the same as the one above. Therefore a cylinder and a plane intersect in an ellipse.
The ice cream cone proof can also be applied to the hyperbola. Tip the plane up so that it intersects both cones. Place a sphere in the upper cone, tangent to the plane, and place a second sphere in the lower cone, tangent to the same plane. The points of tangency are q1 and q2. These become the foci for the hyperbola.
The spheres create an upper ring and a lower ring, as before. One can start at the upper ring and head down, through the apex, and on to the lower ring. This is a fixed distance d.
Let p be a point on the upper branch of the hyperbola, where the plane intersects the top cone. Now the distance from p to q1 is the same as the distance from p to the upper ring. At the same time, the distance from p to q2 is the same as the distance from p, through the apex, to the lower ring. The difference between these distances is d, a constant. Therefore the distance from p to the second focus, minus the distance from p to the first focus, is constant. This defines a hyperbola.