Game of Cones
You're operating a cone-cutting device with the high-energy laser beam. This laser ray is parallel to the sliding belt, and its generator is fixed at all time.
You may simply place a cone on its base or may tilt it a little with a machinery arm. However, if the cone is tilted over the angle (within the cone), it will be toppled to the belt as shown in the picture.
Which conic section is visually impossible to achieve by this machine? (Assume that all the cones stay still enough to be cut.)
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1 solution
Let's suppose that it would be sufficient to create just one branch of a hyperbola (after all, you would need two cones to achieve a hyperbola). In order for this to happen, the axis of the cone must be parallel to the conveyor belt that the cone sits on. Oriented upright, the cone will create a circle. Tilted slightly without tipping, the cone's cross-section will be an ellipse. But if it topples over, the cone's axis will point into the conveyor belt and the cross-sectional cuts will be parabolic.
In order to create the hyperbolic branch desired, the cone would need to be held at an angle that exceeds what is allowed before the cone topples over, so the hyperbola cannot be achieved.
Fun problem!