Mathematics · Ch 5 — Two Dimensional Analytical Geometry-II
Geometric Description of a Conic Section
5.4.1
Geometric Description of a Conic Section
Let the double-napped cone have its axis vertical, and consider a cutting plane that does not pass through the vertex, intersecting only one nappe:
- A plane perpendicular to the axis (plane ) yields a circle.
- A plane tilted, so it is no longer perpendicular to the axis but still crosses only one nappe, yields an ellipse (plane ).
- A plane parallel to a side (generator) of the cone yields a parabola.
- A plane parallel to the axis itself (equivalently, parallel to the plane containing the axis), crossing both nappes, yields a hyperbola — which is why the hyperbola alone has two separate branches: one from each nappe. …
Figure 5.40–5.42Plane sections of a double-napped cone: as the cutting plane's angle changes it yields a circle, an ellipse, a parabola or a hyperbola (schematic)
Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.
What this figure shows. A double cone cut by three differently-tilted planes, tracing out a circle/ellipse pair, a parabola, and a two-branch hyperbola respectively — the geometric definition of a conic section. …