The rear mirrors you see in your car or the huge round silver ones you encounter at a metro station. These mirrors are examples of curves. Curves have huge applications everywhere, be it the study of planetary motion, the design of telescopes, satellites, reflectors etc. Conic sections or conics consist of curves which are obtained upon the intersection of a plane with a double napped right circular cone. Let us discuss the formations of different sections of the cone and their significance.
Consider a fixed vertical line ‘l’ and another line ‘m’ inclined at an angle ‘α’ intersecting l at point V as shown below:
The initials as mentioned in the above figure A carry the following meanings:
- V is vertex of the cone
- l is the axis of the cone
- m, the rotating line the is a generator of the cone
The vertex of the cone divides it into two nappes referred to as the upper nappe and the lower nappe. In figure B, the cone is intersected by a plane and the section so obtained is known as a conic section. Depending upon the position of the plane which intersects the cone and the angle of intersection β, different kind of conic sections are obtained. Namely;
Let us briefly discuss the different conic sections formed when the plane cuts the nappes (excluding the vertex).
- If β=90o, the conic section formed is a circle as shown below.
- If α<β<90o, the conic section so formed is an ellipse as shown in the figure below.
- If α=β, the conic section formed is a parabola (represented by the orange curve) as shown below.
- If 0≤β<α, then the plane intersects both nappes and conic section so formed is known as a hyperbola (represented by the orange curves).
If the plane intersects exactly at the vertex of the cone, following cases may arise:
- If α< β≤90o, then the plane intersects the vertex exactly at a point.
- If α=β, the plane upon an intersection with cone forms a straight line containing a generator of the cone. This condition is a degenerated form of a parabola.
- If 0≤β<α, the section formed is a pair of intersecting straight lines. This condition is a degenerated form of a hyperbola.
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