Concept of conic sections

Concept of conic sections

The term "conic sections" is derived from the words cone and sections. The word sections refers to slices formed when an object is cut at a specific angle.

When a cone is sliced at different angles using a plane, the resulting cross-sections produce different curves. These curves are known as conic sections or simply conics.

A conic section is a curve obtained by the intersection of a plane with a right circular cone at various angles. Depending on the angle and position of the intersecting plane, one of the following four conic sections is formed:

• Circle
• Ellipse
• Parabola
• Hyperbola

Activity 1.1

To better understand conic sections, perform the following activity:

1. Take a soft cone. Use a soft cone made from clay or a paper model.
2. Slice the cone. Slice the cone using a flat plane at different angles.
3. Observe the shapes formed at the cross-sections:
   • If the plane cuts perpendicular to the axis of the cone, the section is a circle.
   • If the plane cuts at an angle, but not parallel to the side of the cone, the section is an ellipse.
   • If the plane is parallel to the side of the cone, the section is a parabola.
   • If the plane cuts through both nappes (the upper and lower parts) of the cone, the section is a hyperbola.

Conic sections and their equations

Each conic section has a standard equation form in Cartesian coordinates:

• Circle: $x^2 + y^2 = r^2$
• Ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
• Parabola: $y^2 = 4ax$ or $x^2 = 4ay$
• Hyperbola: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

These curves have unique properties and appear in many natural and physical phenomena, such as planetary orbits, satellite dishes, and headlights of vehicles.

Different sections of a cone

Hyperbola obtained by sticking cones

It can be observed that:

1. A 90° slice results in a circle.
2. A slice parallel to the generating line creates a parabola.
3. A slice intersecting the cone's axis at an angle less than or greater than 90° forms an ellipse.
4. A slice parallel to the cone's axis intersecting two nappes creates a hyperbola.

In geometry, a conic section is considered a locus of a point. A locus is a set of points whose coordinates are defined by an equation or conditions. A conic section is a locus of a point moving such that its distance from a fixed point (focus) has a constant ratio to its distance from a fixed line (directrix). This chapter focuses on the nature and properties of the parabola, ellipse, and hyperbola.

Properties of conic sections

Conic sections share common properties:

1. Focus: A fixed point.
2. Directrix: A fixed line.
3. A moving point.
4. Center: A point of symmetry (not present in parabolas).
5. Vertex: A turning point.
6. Eccentricity: A constant ratio.

Common terms used in conic sections

Key terms associated with conic sections:

1. Center: The intersection of the major and minor axes (ellipses and hyperbolas only). Parabolas do not have a center.
2. Focus (plural: foci): A fixed point used to define a conic section. Parabolas have one focus, while ellipses and hyperbolas have two.
3. Vertex (plural: vertices): A turning point. Parabolas have one vertex. Ellipses and hyperbolas have two vertices and sometimes co-vertices.
4. Directrix: A line used to define a conic section, related to the focus and eccentricity.
5. Axis (plural: axes): A line of symmetry. Ellipses and hyperbolas have major and minor axes. For hyperbolas, these are called the transverse and conjugate axes, respectively.
6. Latus Rectum: A chord perpendicular to the major axis passing through the focus.

Different geometrical properties of a parabola

Different geometrical properties of an ellipse

Different geometrical properties of a hyperbola

Vertex

A vertex is a turning point. A parabola has one vertex. Ellipses and hyperbolas have two vertices. In an ellipse, the endpoints of the major axis are the vertices, and the endpoints of the minor axis are the co-vertices.

Axis

An axis divides a conic into two equal parts. Ellipses and hyperbolas have major and minor axes. In hyperbolas, the major axis is called the transverse axis, and the minor axis is the conjugate axis.

Center

The center is where the major and minor axes of an ellipse or hyperbola intersect. Parabolas do not have a center.

Focus

A focus is a fixed point used to define a conic section. Parabolas have one focus, while ellipses and hyperbolas have two.

Directrix

The directrix is a line used to define a conic section. The ratio of the distance from a point on the conic to the focus and the distance from that point to the directrix is constant.

Eccentricity

Eccentricity ($e$) is the ratio of the distance from a moving point to the focus and the distance from the moving point to the directrix. It determines the type of conic section:

• $e = 1$: Parabola
• $e < 1$: Ellipse
• $e > 1$: Hyperbola
• $e = 0$: Circle

Latus rectum

The latus rectum is a chord perpendicular to the major axis that passes through the focus.