Conic Sections – Formulas and Diagrams

Conic sections are obtained by the intersection of the surface of a cone with a plane. We can have four types of conic sections that are defined based on the angle formed between the plane and the base of the cone. The four types of conic sections are the circle, ellipse, parabola, and hyperbola.

Here, we will learn more details about each of the different conic sections. We will find out what their formulas are and use diagrams to illustrate them. Finally, we will review the concept of eccentricity, which is an important characteristic of conic sections.

PRECALCULUS
example of conic sections

Relevant for

Learning about the different types of conic sections.

See definition

PRECALCULUS
example of conic sections

Relevant for

Learning about the different types of conic sections.

See definition

What are conic sections?

Conic sections are the curves generated by a plane that intersects a cone. The three types of conic sections are the ellipse, the parabola, and the hyperbola. The circle is a type of ellipse, but it is often considered the fourth type of conic section.

Conic sections are generated by the intersection of a plane with a cone. If the plane is parallel to the axis of revolution (y-axis), then the intersection will form a hyperbola. If the plane is parallel to one side of the cone (generator line), the intersection will form a parabola.

If the plane is perpendicular to the axis of revolution, the intersection will form a circle. If the plane intersects the cone at an angle to the axis, the intersection will form an ellipse.

diagram of the four conic sections

Parameters of a conic section

The following are some of the most important parameters defined in conic sections:

Foci: The foci are two fixed points that define the conic section.

Directrix: The directrix is a straight line that also defines the conic section.

Eccentricity: Eccentricity is a parameter that determines the shape of the conic section. This parameter depends on the length of the semi-major axis and the length of the semi-minor axis.

Focal parameter: The focal parameter is the distance from the focus to the corresponding directrix.

Major axis: Segment that joins the two vertices. In an ellipse, the major axis is the longest diameter.

Minor axis: Segment that joins the covertices. The minor axis is perpendicular to the major axis.


Eccentricity of conic sections

Eccentricity is a parameter that is associated with all conic sections. Eccentricity defines the shape of the conic section and can be thought of as a measure of how much it deviates from being circular.

Eccentricity is defined by c divided by a, where c is the length from the center to the focus and a is the length from the center to the vertex. The value of e is constant for any conic section. Therefore, the value of e can be used to determine the type of conic section:

  • If $latex e=0$, we have a circle
  • If $latex e \leq 0 <1$, we have an ellipse
  • If $latex e = 1$, we have a parabola
  • If $latex e> 1$, we have a hyperbola

A circle is defined as a special type of an ellipse with an eccentricity of 0. Two conic sections have the same shape only if their eccentricity is the same.


Conic section – Circle

Circles are formed when the plane that intersects the cone is parallel to the base of the cone. The intersection produces a set of points that have the same distance from a common point, which is the definition of a circle.

All circles have a central point, called the center, and a radius, which is the distance from the center to any point on the circle. Also, the circles have an eccentricity of $latex e = 0$. In the Cartesian plane, the general form of the equation of the circle is:

$latex {{(x-h)}^2}+{{(y-k)}^2}={{r}^2}$

where, $latex (h, k)$ is the center of the circle and r is the radius.

diagram of a circumference

Conic Section – Ellipse

Ellipses are obtained when the angle of the plane relative to the cone is between the outer surface of the cone and the base of the cone. This definition also includes the case where the plane is parallel to the base of the cone, so circles are a special case of ellipses.

Ellipses have the following characteristics:

  • The major axis is the longest diameter of the ellipse
  • The minor axis is the shortest diameter of the ellipse
  • The center is the intersection of the two centers
  • They have two foci. The sum of the distances from any point on the ellipse to the two foci is constant

Ellipses can have the eccentricity $latex 0 \leq e <1$. We see that the value of 0 is included (a circle), but 1 is not included since it is the eccentricity of a parabola. When the major axis is parallel to the x axis, the general equation of an ellipse is:

$latex \frac{{{(x-h)}^2}}{{{a}^2}}+\frac{{{(y-k)}^2}}{{{b}^2}}=1$

where, $latex (h, k)$ is the center, 2a is the length of the major axis, and 2b is the length of the minor axis.

Parameters of horizontal ellipses centered outside the origin

Conic Section – Parabola

The parabolas are formed when the plane is parallel to the sides of the cone. This results in a “U” shaped curve. The parabolas have the following characteristics:

  • The vertex is the point where the curve changes direction
  • The focus is the point that is in the internal part of the parabola and the one that gives the shape to the curve
  • The directrix is the line that is on the outside of the parabola and that also defines it
  • The axis of symmetry is the line that connects the vertex and the focus and divides the parabola into two equal parts

The parabolas have the eccentricity $latex e = 1$. Since all parabolas have the same eccentricity, they have very similar characteristics and can be transformed with a change of position and scaling.

Parabolas can be represented by quadratic functions like:

$latex f(x)={{x}^2}$
vertex form of a parabola

Conic Section – Hyperbola

Hyperbolas are formed when the plane is parallel to the central axis of the cone. This means that the plane intersects both bases of the cone. Hyperbolas are composed of two branches and have the following characteristics:

  • Asymptotes are two straight lines that the curve approaches, but never touches
  • The center is the intersection of the two asymptotes
  • The two foci are the fixed points, which define the shape of each branch
  • The two vertices are the points that are located one on each branch and where each branch changes direction

The eccentricity of the hyperbola is equal to $latex e> 1$. A hyperbola that has the vertices on a horizontal line has the general equation:

$latex \frac{{{(x-h}^2}}{{{a}^2}}-\frac{{{(y-k)}^2}}{{{b}^2}}=1$

where, $latex (h, k)$ is the center, $latex 2a$ is the length of the segment connecting the vertices, and $latex 2b$ is the length of the segment connecting the covertices.

hyperbola oriented vertically with center outside the origin

Formulas for conic sections

The following are the formulas for the different types of conic sections:

Circle$latex {{(x-h)}^2}+{{(y-k)}^2}={{r}^2}$Center: (h, k)Radius: r
Horizontal ellipse$latex \frac{{{(x-h)}^2}}{{{a}^2}}+\frac{{{(y-k)}^2}}{{{b}^2}}=1$Center: (h, k)
Length of major axis: 2a
Length of minor axis: 2b
Vertical ellipse$latex \frac{{{(x-h)}^2}}{{{b}^2}}+\frac{{{(y-k)}^2}}{{{a}^2}}=1$Center: (h, k)
Length of major axis: 2a
Length of minor axis: 2b
Horizontal parabola$latex {{(y-k)}^2}=4p(x-h)$Vertex: (h, k)
Focus: (h+p, k)
Vertical parabola$latex {{(x-h)}^2}=4p(y-k)$Vertex: (h, k)
Focus: (h, k+p)
Horizontal hyperbola$latex \frac{{{(x-h)}^2}}{{{a}^2}}-\frac{{{(y-k)}^2}}{{{b}^2}}=1$Center: (h, k)
Distance between vertices: 2a
Distance between covertices: 2b
Vertical hyperbola$latex \frac{{{(y-k)}^2}}{{{a}^2}}+\frac{{{(x-h)}^2}}{{{b}^2}}=1$Center: (h, k)
Distance between vertices: 2a
Distance between covertices: 2b
Circle$latex {{(x-h)}^2}+{{(y-k)}^2}={{r}^2}$
Horizontal ellipse$latex \frac{{{(x-h)}^2}}{{{a}^2}}+\frac{{{(y-k)}^2}}{{{b}^2}}=1$
Vertical ellipse$latex \frac{{{(x-h)}^2}}{{{b}^2}}+\frac{{{(y-k)}^2}}{{{a}^2}}=1$
Horizontal parabola$latex {{(y-k)}^2}=4p(x-h)$
Vertical parabola$latex {{(x-h)}^2}=4p(y-k)$
Horizontal hyperbola$latex \frac{{{(x-h)}^2}}{{{a}^2}}-\frac{{{(y-k)}^2}}{{{b}^2}}=1$
Vertical hyperbola$latex \frac{{{(y-k)}^2}}{{{a}^2}}+\frac{{{(x-h)}^2}}{{{b}^2}}=1$

See also

Interested in learning more about conic sections? Take a look at these pages:

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