Conic Sections

A conic section, also called conic in geometry is formed when a plane intersects a cone at different angles and positions. It can be a circle, ellipse, parabola, or hyperbola according to the varied angles of intersection.

Architectural designs like bridges, towers, and buildings that are circle, ellipse, parabola, or hyperbola-shaped are real-life examples of conic sections.

Types

Circle 

It is formed when a plane cuts a cone parallel to its base. It is perpendicular to the axis of revolution, marked with the red dot.

Ellipse 

It is formed when a plane cuts a cone at an angle. 

Parabola 

It formed when a plane cuts a cone parallel to its slant height. The slant height is also considered the generator line as it is the outer surface of a cone.

Hyperbola 

It is formed when a plane cuts a cone parallel to its axis of revolution.

Parameters

Some important terms related to conical sections are:

Focus

The focus is the point about which a conic section is created. It lies on the major axis.

• Ellipses and hyperbolas have 2 foci
• Parabolas have 1 focus
• Circle has 2 foci both at the same point: the center

It is denoted by F. When we draw two foci for a conic, F₁ and F₂ mark the extremities.

Latus Rectum

It is parallel to the directrix of a conic section and passes through the focus.

• For a parabola, it is four times the focal length
• For a circle, it is the diameter
• For an ellipse, it is 2b²/a (where a = semi-major axis, b = semi-minor axis). Its length is generally denoted as LL’.

Directrix

It is the line parallel to the conjugate axis and the latus rectum.

Eccentricity

The eccentricity determines the round or flatness of a conic section. It is denoted by e.

• For a circle, e = 0,
• For an ellipse, e < 1,
• For a parabola, e = 1, and
• For a hyperbola, e > 1

Other than the above parameters, some more terms related to conic section are:

Principal axis also called the major axis, it is the axis passing through the center and foci of a conic.

Conjugate Axis also called the minor axis, it is the axis drawn perpendicular to the principal axis and passing through the center of the conic.

Center is the point of intersection of the principal axis and the conjugate axis of the conic.

Vertex is the point where the conic cuts the axis.

Focal Chord is the chord passing through the focus of the conic.

Focal Distance is the distance of a point (x₁, y₁) on the conic, from any of the foci

Asymptotes is a pair of straight lines drawn parallel to the hyperbola which is assumed to touch the hyperbola at infinity.

Equations

The general equation of a conic section is:

Ax² + Bxy + Cy² + Dx + Ey + F = 0, here A, B, C, D, E, and F are constants

From the above equation we can derive specific equations for the circle, ellipse, parabola and hyperbola.

Circle

r² = (x-h)² + (y-k)², here r = radius, (h, k) = center, (x, y) = a point on the circle

Ellipse

${\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1}$, where a = semi-major axis, b = semi-minor axis, (x, y) = a point on the ellipse

Parabola

4py = x², here p = distance from vertex to focus (or directrix), (x, y) = a point on the parabola

Hyperbola

${\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1}$, where a = semi-major axis, b = semi-minor axis, (x, y) = a point on the hyperbola

How to Identify the Conic Section

The steps to identify the graph of a conic section from its general form of the equation are:

1. If A and C are equal and non zero, and both have the same sign, then it is a circle
2. If A and C are unequal and non zero, and have the same sign, then it is an ellipse
3. If A or C is zero, then it is a parabola
4. If A and C are non zero and have different signs, then it will be a hyperbola