Conic sections: slicing a cone
Cut a cone with different planes and see where the circle, ellipse, parabola and hyperbola come from. Then see what the ratio of the distance to a focus and to a directrix builds.
- Iran: grade 12
Guide
This studio follows the Iranian Year 12 geometry 3 lesson on conic sections and the spatial-thinking lesson of Mathematics 3. If a plane cuts a cone, the edge of the cut is a curve. These curves are the conic sections.
A cut parallel to the base gives a circle. Tilt the plane, but less steeply than the cone's side, and you get an ellipse. A plane parallel to a side of the cone gives a parabola, and a steeper plane gives a hyperbola.
Each curve is also a locus: the points whose distance from a fixed point (the focus) divided by their distance from a fixed line (the directrix) is a constant. That constant is the eccentricity, e. Common mistakes: thinking only a flat cut makes a circle and the others are something else, and turning the ratio upside down when finding e.