UNIVERSITY OF THE EAST

Introduction to Conic Sections

A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes called a fourth type.

A conic section is the locus of a point moving in a plane, such that its distance from a fixed point (focus) is in a constant ratio to its perpendicular distance from a fixed line (i.e. directrix). Conic sections can be obtained as intersections of a plane with a double-napped right circular cone.

General Form of Conic Section

The General form of the equation of conic section is denoted by:

Ax2 +Bxy+Cy2 +Dx+Ey+F=0

Where: A, B, C, D, E, & F are constants and real numbers.

The applications of these curves include the design of telescopes, automobile headlights and reflectors in flashlights, etc. The constant ratio is known as the eccentricity of the conic.

In this article, we discuss how to identify conic sections from the general form.

CIRCLE

Circle: It is the set of all points in a plane that are equidistant from a fixed point in the plane. The center of the circle is the fixed point. The fixed distance from the center to any point on the circle is called the radius.

Standard equation is given by (x-h)2 + (y-k)2 = r2. Centre is (h, k) and radius is r.
Standard equation is given by x2 +y2 =r2 . Center is at origin and radius is r.

ELLIPSE

Ellipse: It is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is constant. The two fixed points are called the foci of the ellipse.

Equation is given by x2/a2+y2/b2=1. Center is at origin with the x-axis as its major axis
Equation is given by x2/b2+y2/a2=1. Center is at origin with the y-axis as its major axis.

Equation is given by (x-h)2/a2 + (y-k)2/b2 = 1. Center is at (h, k) with the x-axis as its major axis.
Equation is given by (x-h)2/b2 + (y-k)2/a2 = 1. Center is at (h, k) with the y-axis as its major axis.

PARABOLA

Parabola: It is the set of all points in a plane that are equidistant from a fixed line and a fixed point (not on the line) in the plane. The fixed-line is called the directrix of the parabola and the fixed point F is called the focus.

Equation of a parabola is y2=4px. Vertex is at origin , horizontal axis of symmetry.
Equation of a parabola is x2=4py. Vertex is at origin , vertical axis of symmetry.

Equation of a parabola is (y-k)2 = 4p(x-h). Vertex is at (h, k) , horizontal axis of symmetry.
Equation of a parabola is (x-h)2 = 4p(y-k). Vertex is at (h, k) , vertical axis of symmetry.

HYPERBOLA

Hyperbola: It is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is constant.

Equation of hyperbola is x2/a2-y2/b2=1. When transverse axis is on the x-axis.
Equation of hyperbola is x2/b2-y2/a2=1. When transverse axis is on the y-axis.

Equation of hyperbola is (x-h)2/a2 – (y-k)2/b2 = 1. When the transverse axis is parallel to the x-axis.
Equation of hyperbola is (x-h)2/b2 – (y-k)2/a2 = 1. When the transverse axis is parallel to the y-axis.

Steps to Identify Conic Sections From General Form

1. If A and C are non zero and equal, and both have the same sign, then it will be a circle.

2. If A and C are non zero and unequal, and have the same sign, then it will be an ellipse.

3. If A or C is zero, then it will be a parabola.

4. If A and C have different signs and are non zero, then it will be a hyperbola.

Solved Example: Solution:

Example: Identify the graph of following conic section Given 4x2 – 25y2 – 24x + 250 y – 489 = 0

4x2 – 25y2 – 24x + 250y – 489 = 0 Here A = 4 and C =-25

a) circle A and C have different signs and non zero. So it is a b) Hyperbola hyperbola.

c) Ellipse Hence option b) is the answer.

d) Parabola

APPLICATIONS

circle.mp4

hyperbola.mp4

REFERENCES

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