Conic Sections

Conic Sections: An Introduction

Conic sections (or simply conic) are curves obtained as the intersection of the surface of a cone with a plane; the three types are parabolas, ellipses, and hyperbolas.

  • A conic section can be graphed on a coordinate plane.
  • Every conic section has certain features, including at least one focus and directrix. Parabolas have one focus and directrix, while ellipses and hyperbolas have two of each.
  • A conic section is the set of points P whose
    distance to the focus is a constant multiple of the distance from P to the directrix of the conic.

Key terms used in Conic Sections

  • Vertex: An extreme point on a conic section.
  • Asymptote: A straight line which a curve approaches arbitrarily closely as it goes to infinity.
  • Locus: The set of all points whose coordinates satisfy a given equation or condition.
  • Focus: A point used to construct and define a conic section, at which rays reflected from the curve converge (plural: focii).
  • Nappe: One half of a double cone.
  • Conic Section: Any curve formed by the intersection of a plane with a cone of two nappes.
  • Directrix: A line used to construct and define a conic section; a parabola has one directrix; ellipses and hyperbolas have two (plural: directrices).

Parts of a Conic Section

While each type of conic section looks very different, they have some features in common. For example, each type has at least one focus and directrix.

A focus is a point about which the conic section is constructed. In other words, it is a point about which rays reflected from the curve converge. A parabola has one focus about which the shape is constructed; an ellipse and hyperbola have two.

A directrix is a line used to construct and define a conic section. The distance of a directrix from a point on the conic section has a constant ratio to the distance from that point to the focus. As with the focus, a parabola has one directrix, while ellipses and hyperbolas have two.

These properties that the conic sections share are often presented as the following definition, which will be developed further in the following section. A conic section is the locus of points P whose distance to the focus is a constant multiple of the distance from P to the directrix of the conic. These distances are displayed as orange lines for each conic section in the following diagram.

Different Conic Sections

Circle:

A circle is formed when the plane is parallel to the base of the cone. Its intersection with the cone is therefore a set of points equidistant from a common point (the central axis of the cone), which meets the definition of a circle. All circles have certain features:

  • A center point
  • A radius, which the distance from any point on the circle to the center point

All circles have an eccentricity . Thus, like the parabola, all circles are similar and can be transformed into one another. On a coordinate plane, the general form of the equation of the circle is

where  are the coordinates of the center of the circle, and  is the radius.

The degenerate form of the circle occurs when the plane only intersects the very tip of the cone. This is a single point intersection, or equivalently a circle of zero radius.

 

Parabola:

A parabola is the set of all points whose distance from a fixed point, called the focus, is equal to the distance from a fixed line, called the directrix. The point halfway between the focus and the directrix is called the vertex of the parabola.

The following shows the general equation of a Parabola:

In the next figure, four parabolas are graphed as they appear on the coordinate plane. They may open up, down, to the left, or to the right.

Ellipse:

An ellipse is the set of all points for which the sum of the distances from two fixed points (the foci) is constant. In the case of an ellipse, there are two foci, and two directrices.

Ellipses can have a range of eccentricity values: . Notice that the value  is included (a circle), but the value  is not included (that would be a parabola). Since there is a range of eccentricity values, not all ellipses are similar. The general form of the equation of an ellipse with major axis parallel to the x-axis is:

where  are the coordinates of the center,  is the length of the major axis, and  is the length of the minor axis. If the ellipse has a vertical major axis, the  and b labels will switch places.

The degenerate form of an ellipse is a point, or circle of zero radius, just as it was for the circle.

In the next figure, a typical ellipse is graphed as it appears on the coordinate plane.

Hyperbola:

A hyperbola is the set of all points where the difference between their distances from two fixed points (the foci) is constant. In the case of a hyperbola, there are two foci and two directrices. Hyperbolas also have two asymptotes.

The general equation for a hyperbola with vertices on a horizontal line is:

where  are the coordinates of the center. Unlike an ellipse,  is not necessarily the larger axis number. It is the axis length connecting the two vertices.

The eccentricity of a hyperbola is restricted to , and has no upper bound. If the eccentricity is allowed to go to the limit of (positive infinity), the hyperbola becomes one of its degenerate cases — a straight line. The other degenerate case for a hyperbola is to become its two straight-line asymptotes. This happens when the plane intersects the apex of the double cone.

A graph of a typical hyperbola appears in the next figure.

 

 

Eccentricity of a Conic Section (e)

The eccentricity, denoted e, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.

The eccentricity of a conic section is defined to be the distance from any point on the conic section to its focus, divided by the perpendicular distance from that point to the nearest directrix. The value of e is constant for any conic section. This property can be used as a general definition for conic sections. The value of  can be used to determine the type of conic section as well:

  • If , the conic is a parabola
  • If , it is an ellipse
  • If , it is a hyperbola

The eccentricity of a circle is zero. Note that two conic sections are similar (identically shaped) if and only if they have the same eccentricity.

Note: The eccentricity of a circle is zero.

Eccentricity is the ratio between the distance from any point on the conic section to its focus, and the perpendicular distance from that point to the nearest directrix.

 

Formula Revision

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Anurag Bansal
Anurag Bansal

Anurag is a Computer Science Engineering student from RVCE Bangalore. Apart from writing for Toppr.com, he has his own Blogging platform called Chronicle Scripts (http://chroniclescripts.com/) A start-up enthusiast, he loves studying business models.

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