Conic Sections

Conic sections or sections of a cone are the curves obtained by the intersection of a plane and cone. In the following guide, you will learn more about the types of conic.

Conic Sections

There are three major sections of a cone or conic sections: parabola, hyperbola, and ellipse (the circle is a special kind of ellipse). A cone with two identical nappes is used to produce the conic sections.

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Step by Step guide to a conic section

Conic sections are the curves obtained when a plane cuts the cone. A cone generally has two identical conical shapes known as nappes. We can get various shapes depending upon the angle of the cut between the plane and the cone and its nappe. By cutting a cone with a plate at different angles, we get the following shapes:

  • Ellipse: An ellipse is a conic section that forms when a plane intersects with a cone at an angle.
  • Circle: A circle is a special type of ellipse in which the cutting plane is parallel to the base of the cone.
  • Hyperbola: A hyperbola is formed when the interesting plane is parallel to the axis of the cone and intersects with both the nappes of the double cone.
  • Parabola: When the intersecting plane cuts at an angle to the surface of the cone, we obtain a conic section called a parabola.

Conic section parameters

Focus

The focus or foci (plural) of a conic section is the point (s) about which the conic section is created. They are specially defined for each type of conic section. A parabola has one focus, while ellipses and hyperbolas have two foci. For an ellipse, the total distance of a point on the ellipse from the two foci is constant. A circle, which is a special state of an ellipse, has both foci at the same place and the distance of all points from the focus is constant.

For parabola, it is a limiting case of an ellipse and has one focus at a distance from the vertex, and another focus at infinity. The hyperbola has two foci and the absolute difference of the distance of the point on the hyperbola from the two foci is constant.

Directrix

Directrix is a line used to define conic sections. A directrix is a line that is perpendicular to the axis of the cone. Each point on a cone is defined by the ratio of its distance from directrix to focus. The directrix is parallel to the conjugate axis and the latus rectum of the conic.

The circle has no directrix. A parabola has \(1\) directrix, ellipse, and hyperbolic have \(2\) directions each.

Eccentricity

The eccentricity of a conic section is the constant ratio of the distance of the point on the conic section from the focus and directrix. Eccentricity is used to uniquely define the shape of a conic section.  It is a non-negative real number. Eccentricity is denoted by “e”.

If two conic sections have the same eccentricity, they will be the same. As eccentricity increases, the conic section deviates more and more from the shape of the circle. The value of \(e\) for different conic sections is as follows:

  • For circle, \(e = 0\).
  • For ellipse, \(0 ≤ e < 1\)
  • For parabola, \(e = 1\)
  • For hyperbola, \(e > 1\)   

Terms related to conic section

The following are the details of the parameters of the conic section:

Principal Axis:

The axis that is perpendicular to the principal axis and passes through the center of the conic is the conjugate axis. The conjugate axis is also its minor axis.

Center:

The point of intersection of the principal axis and the conjugate axis of the conic is called the center of the conic.

Vertex:

The point on the axis that the conic cuts the axis is called the vertex of the conic.

Focal Chord:

The focal chord of a conic is the chord passing through the focus of the conic section. The focal chord cuts the conic section at two distinct points.

Focal Distance:

The distance of a point \((x_1,y_1)\) on the conic, from any of the foci, is the focal distance. For an ellipse, hyperbola we have two foci, and hence we have two focal distances.

Latus Rectum:

It is a focal chord that is perpendicular to the axis of the conic. The length of the latus rectum for a parabola is \(LL’=4a\). The length of the latus rectum for an ellipse and hyperbola is \(\frac {2b^2}{a}\).

Tangent:

The tangent is a line touching the conic externally at one point on the conic. The point of contact with the conic is called the point of contact. Also from an external point, about two tangents can be drawn to the conic.

Normal:

The line is drawn perpendicular to the tangent and passes through the point of contact and the focus of the conic is called the normal.

Chord of Contact:

The chord drawn to join the point of contact of the tangents, drawn from an external point to the conic is called the chord of contact.

Director Circle: 

The locus of the point of intersection of the perpendicular tangents drawn to the ellipse is called the director circle. For an ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\), the equation of the director circle is \(x^2+y^2=a^2+b^2\).

Asymptotes:

The pair of straight lines are drawn parallel to the hyperbola and assumed to touch the hyperbola at infinity. The equations of the asymptotes of the hyperbola are \(y=\frac{bx}{a}\), and \(y=-\frac{bx}{a}\) respectively.

for a hyperbola having the conic equation of \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\), the equation of the pair of asymptotes of the hyperbola is \(\frac{x}{a}±\frac{y}{b}=0\).

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