How to Identify Conic Sections from General Form

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. 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. The eccentricity of a circle is zero. It shows how “un-circular” a curve is. The larger the eccentricity, the lower curved it is. In this article, we discuss how to identify conic sections from the general form.

Different Conic Sections

1. Circle: It is the set of all points in a plane that are equidistant from a fixed point in the plane. The centre of the circle is the fixed point. The fixed distance from the centre 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.

2. 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 (y-k)² = 4p(x-h).

3. 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 (x-h)²/a² – (y-k)²/b² = 1.

4. 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 (x-h)²/a² + (y-k)²/b² = 1.

Conic Section Videos

Visualising Conic Sections

Steps to Identify Conic Sections From General Form

The general equation for any conic section is Ax²+Bxy+Cy²+Dx+Ey+F = 0, where A, B, C, D, E, F are constants.

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

Example: Identify the graph of following conic section

4x² – 25y² – 24x + 250y – 489 = 0

(1) circle

(2) Hyperbola

(3) Ellipse

(4) Parabola

Solution:

Given 4x² – 25y² – 24x + 250 y – 489 = 0

Here A = 4 and C =-25

A and C have different signs and non zero. So it is a hyperbola.

Hence option (2) is the answer.