Conic sections
Introduction to Conic Sections
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Conic sections or conics are the curves obtained by intersecting a double-napped right-circular cone with a plane.
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The concept of conic sections is widely used in astronomy, projectile motion of an object, etc.
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The example of conic sections are circle (Figure I), ellipse (Figure II), parabola (Figure III) and hyperbola (Figure IV).
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Different types of conics can be formed by intersecting a plane with a double-napped cone (other than the vertex) by different ways.
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If θ1 is the angle between the axis and the generator and θ2 is the angle between the plane and the axis, then for different conditions of θ1 and θ2, we get different conics. These are described in the table shown below.
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Condition
Conic Formed
Figure
θ2 = 90° (The plane cuts only one nappe of the cone entirely)
Circle
θ1 < θ2 < 90° (The plane cuts only one nappe of the cone entirely)
Ellipse
θ1 = θ2 (The plane cuts only one nappe of the cone entirely)
Parabola
0 ≤ θ2 < θ1 (The plane cuts each nappe of the cone entirely)
Hyperbola
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The conic sections obtained by cutting a plane with a double-napped cone at its vertex are known as degenerated conic sections.
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If θ1 is the angle between the axis and the generator and θ2 is the angle between the plane and the axis, then for different conditions of θ1 and θ2, we get different conics. These are described in the table shown below.
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Condition
Conic Formed
Figure
θ1 < θ2 ≤ 90°
Point
θ1 =θ2
Line
0 ≤ θ2 < θ1
Hyperbola
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Key Concepts
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A parabola is defined as the set of all points in a plane that are equidistant from a fixed line and a fixed point (not on the line) in the plane.
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If the fixed point lies on the fixed-line, then the resulting set of points is a straight line. This straight line passes through the fixed point and is perpendicular to the fixed-line. This is the degenerated case of a parabola.
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Some of the terms associated with a parabola are directrix,…
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