Table of Contents

Last modified on August 3rd, 2023

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.

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.

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

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.

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

Some important terms related to conical sections are:

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, F_{1} and F_{2} mark the extremities.

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 2b
^{2}/a (where a = semi-major axis, b = semi-minor axis). Its length is generally denoted as LL’.

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

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 (x_{1}, y_{1}) 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.

The general equation of a conic section is:

Ax^{2} + Bxy + Cy^{2} + 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.

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

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

4py = x^{2}, here p = distance from vertex to focus (or directrix), (x, y) = a point on the parabola

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

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

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

**Identify the graph of the following equation in standard form.****4x ^{2} – 25y^{2 }– 24x + 250y – 489 = 0**

Solution:

Given 4x^{2} – 25y^{2} – 24x + 250 y – 489 = 0

Here A = 4 and C =-25

Since, both A and C have different signs and are non zeros, it is a hyperbola.

Last modified on August 3rd, 2023