MATHPOWER TM 12, WESTERN EDITION Chapter 3 Conics 3.1

3.1.2 The Double-Napped Cone A Greek mathematician, Appollonius was the first to discover the shapes that occur when a plane intersects the double-napped cone. He pictured the cone as two that are connected at their respective vertices and extend away infinitely.

The Double-Napped Cone [cont’d] The nappes may be developed by rotating an oblique line about a vertical axis. This oblique line is referred to as the Double-Napped Cone Upper Nappe Lower Nappe Vertex Generator Axis of symmetry

3.1.4 The Circle When a double-napped cone is intersected by a plane at right angles to its axis but not through the vertex, the cross section is a When the cutting plane is moved, the circle becomes When the plane moves the circle becomes

If the plane intersects one nappe of the double-napped cone but at right angles to the axis, parallel to the generator, and through the vertex, the cross-section is an The Ellipse As the cutting plane cone, the ellipse becomes 3.1.5

3.1.6 If the plane intersects one nappe of the double-napped cone parallel to the generator but, the cross section is a parabola. The Parabola As the cutting plane of the cone, the parabola becomes

3.1.7 The Hyperbola If the plane intersects both nappes of the double-napped cone but not through the vertex, the cross-section is a As the cutting plane, the branches of the hyperbola become

When a double-napped cone is intersected by a plane at right angles to the axis and through the vertex, the cross-section is a. The Degenerate Conic Sections Thus, for the circle, when the plane intersects with the vertex, the degenerate conic is the point

The Degenerate Conic Sections If the plane intersects the double-napped cone through the vertex but not at right angles to the axis and not parallel to the generator, the cross-section is a point. Thus, the degenerate of the ellipse is a point

The Degenerate Conic Sections If the plane intersects the double-napped cone parallel to the generator and through the vertex, the cross-section is a line. Thus, the degenerate of the parabola is a line

The Degenerate Conic Sections If the plane intersects the double-napped cone through the vertex but not parallel to the generator, the cross-section is two intersecting lines. Thus, the degenerate of the hyperbola is intersecting lines

If the generator is made parallel to the axis and then rotated about the axis, a cylinder results. The Cylinder Generator

The Cylinder

Circle: Ellipse: Parabola: Hyperbola: Finding the Angle Through Which Conic Sections are Formed With reference to the horizontal axis:

The Cutting Plane and the Double-Napped Cone 1. A glass of red juice is tilted but not to the point of spilling. Which conic section describes the shape the juice in the glass? 2. A plane intersects a double-napped cone at a point parallel to the axis of the cone but excluding the vertex. Describe the conic section produced. 3. When a horizontal plane cuts a double-napped cone at the vertex, which conic section is produced? 4. The equation 4x + 3y = 12 produces the degenerate of which conic?

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