Conic sections are curves obtained by intersecting a plane and cone, consisting of three major sections: parabola, hyperbola, and ellipse. These sections share some common properties, such as their shape and shape. The cone with two identical nappes is used to produce conic sections.

The three types of conic sections are the parabola, the parabola, and the ellipse, with the circle being a special case of the ellipse. Ancient Greek mathematicians studied conic sections, culminating in Apollonius of Perga’s systematic work on their properties around 200 BC. Well, in this reading, we’ll explore what a conic section is, its definition, applications, formulas, equations, examples, and parameters.

Let’s get started!

**What is a Conic Section?**

Conic sections are curves generated by intersecting a right circular cone with a plane in Euclidean geometry. These sections have distinct properties, with the cone’s vertex dividing it into two nappes: upper and lower. The position of the plane intersecting the cone and the angle of intersection β determine the type of conic section.

Examples include circles, ellipses, parabolas, and hyperbolas. Curves have significant applications in various fields, such as studying planetary motion and designing telescopes, satellites, and reflectors. Conic sections consist of curves obtained upon the intersection of a plane with a double-napped right circular cone.

Understanding the formation of different sections of the cone and their significance is essential for understanding their applications.

**Definition**

A conic is a curve formed by the intersection of a plane, called the cutting plane, with the surface of a double cone. Conics can be right circular or non-degenerate, with planes passing through the cone intersecting in points, lines, or pairs.

There are three types of conics: ellipses, parabolas, and hyperbolas. The circle is a special type of ellipse, with the cutting plane being parallel to the plane of the generating circle of the cone. Ellipses arise when the intersection of the cone and plane is a closed curve.

If the cutting plane is parallel to exactly one generating line of the cone, the conic is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola, with the plane intersecting both halves of the cone, producing two separate unbounded curves.

**Applications of Conic Sections**

Conic sections, based on Newton’s law of universal gravitation, are used in astronomy to design searchlights, radio telescopes, and optical telescopes. These sections are formed when two massive objects have a common center of mass at rest. The reflective properties of conic sections are utilized in the design of these sections, such as searchlights and microphones.

A parabolic mirror serves as the reflector in searchlights, while a parabolic microphone is used in parabolic microphones. The Herschel optical telescope on La Palma uses a primary parabolic mirror to reflect light towards a secondary hyperbolic mirror.

**Formulas of Conic Sections**

The table below shows the various formular of conic sections:

Circle | (x−a)^{2}+(y−b)^{2}=r^{2} |
Center is (a,b)
Radius is r |

Ellipse with the horizontal major axis | (x−a)^{2}/h^{2}+(y−b)^{2}/k^{2}=1 |
Center is (a, b) Length of the major axis is 2h. Length of the minor axis is 2k. Distance between the centre and either focus is c with c ^{2}=h^{2}−k^{2}, h>k>0 |

Ellipse with the vertical major axis | (x−a)^{2}/k^{2}+(y−b)^{2}/h^{2}=1 |
Center is (a, b) Length of the major axis is 2h. Length of the minor axis is 2k. Distance between the centre and either focus is c with c ^{2}=h^{2}−k^{2}, h>k>0 |

Hyperbola with the horizontal transverse axis | (x−a)^{2}/h^{2}−(y−b)^{2}/k^{2}=1 |
Center is (a,b) Distance between the vertices is 2h Distance between the foci is 2k. c ^{2}=h^{2} + k^{2} |

Hyperbola with the vertical transverse axis | (x−a)^{2}/k^{2}−(y−b)^{2}/h^{2}=1 |
Center is (a,b) Distance between the vertices is 2h Distance between the foci is 2k. c ^{2}= h^{2} + k^{2} |

Parabola with the horizontal axis | (y−b)^{2}=4p(x−a), p≠0 |
Vertex is (a,b) Focus is (a+p,b) Directrix is the line x=a−p Axis is the line y=b |

Parabola with vertical axis | (x−a)^{2}=4p(y−b), p≠0 |
Vertex is (a,b) Focus is (a+p,b) Directrix is the line x=b−p Axis is the line x=a |

**Conic Sections Equations**

Conic section Name | Equation when the centre is at the Origin, i.e. (0, 0) | Equation when centre is (h, k) |

Circle | x^{2} + y^{2} = r^{2}; r is the radius |
(x – h)^{2} + (y – k)^{2} = r^{2}; r is the radius |

Ellipse | (x^{2}/a^{2}) + (y^{2}/b^{2}) = 1 |
(x – h)^{2}/a^{2} + (y – k)^{2}/b^{2} = 1 |

Hyperbola | (x^{2}/a^{2}) – (y^{2}/b^{2}) = 1 |
(x – h)^{2}/a^{2} – (y – k)^{2}/b^{2} = 1 |

Parabola | y^{2} = 4ax, where a is the distance from the origin to the focus |

**Conic Parameters**

Conic sections are defined by parameters such as the principal axis, linear eccentricity, latus rectum, focal parameter, major axis, and minor axis. These parameters include focus, eccentricity, directrix, and chord length. The principal axis is the line joining the two focal points, while the major axis is the longest chord of an ellipse, and the minor axis is the shortest chord.

**Focus, Eccentricity and Directrix of Conic**

A conic section is a plane with a fixed point F and a fixed line d. The locus of a point P moving in the plane is known as focus (F) and the distance from d is known as the directrix (d). Eccentricity measures the deviation of the ellipse from being circular.

If eccentricity is 0, the conic is a circle, otherwise, it is a hyperbola. The eccentricity is calculated by dividing the angle between the cone’s surface and its axis by the angle between the cutting plane and the axis.

So, eccentricity is a measure of the deviation of the ellipse from being circular. Suppose, the angle formed between the surface of the cone and its axis is β and the angle formed between the cutting plane and the axis is α, the eccentricity is:

e = cos α/cos β

**Conic Section of Circle, Parabola, Ellipse, and Hyperbola**

The circle is a special type of ellipse with a focus at its center. The radius of the circle is the fixed distance from the center, and the eccentricity (e) is equal to zero. The section of the circle is parallel to the base of the cone, forming a conic section.

A parabola is a U-shaped conic section formed by the intersection of a cone with a parallel plane. It is an asymmetrical open plane curve with an eccentricity of 1. The graph of a quadratic function is a parabola, a line-symmetric curve with a shape similar to the graph of y = x2. The path of a projectile under gravity ideally follows a curve of this shape, allowing it to open upward or downward.

An ellipse is a conic section formed when a plane intersects with a cone at an angle. It has two foci, a major axis, and a minor axis. The value of eccentricity is e < 1, and the equation has a center at (h, k) and lengths of ‘2a’ and ‘2b’ respectively.

A hyperbola is a conic section formed when the plane intersects with the axis of a double cone, with a value of eccentricity (e) > 1. The two unconnected sections, called branches, are mirror images of each other and approach the limit to a line. A hyperbola is an example of a conic section drawn on a plane.

**Terms Used in Conic Section**

Conic sections are a type of geometry that involves the use of straight lines to create a conic section. These sections are characterized by their principal axis, which is the axis passing through the center and foci of the conic.

The conjugate axis is the axis drawn perpendicular to the principal axis and passing through the center of the conic.

The center is the point of intersection of the principal axis and the conjugate axis, while the vertex is the point on the axis where the conic cuts the axis.

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

The focal distance is the distance from any of the foci on the conic, with two focal distances for an ellipse or hyperbola.

The latus rectum is a perpendicular chord perpendicular to the axis of the conic, with the length of LL’ = 4a and 2b2/a for an ellipse and hyperbola.

A tangent is a line touching the conic externally at one point on the conic, known as the point of contact.

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

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

An auxiliary circle is a circle drawn on the major axis of the ellipse as its diameter, with the conic equation of an ellipse being x2/a2 + y2/b2 = 1.

The director circle is the locus of the point of intersection of the perpendicular tangents drawn to the ellipse.

Asymptotes are the pair of straight lines drawn parallel to the hyperbola and assumed to touch it at infinity.