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.
- إيران: الصف ١٢
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الدليل
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 مخروط, the edge of the cut is a curve. These curves are the القطوع المخروطية.
A cut parallel to the base gives a دائرة. Tilt the plane, but less steeply than the cone's side, and you get an قطع ناقص. A plane parallel to a side of the cone gives a قطع مكافئ, and a steeper plane gives a قطع زائد.
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.