Conic sections

Generation of Conics: An Introduction

Conic sections in mathematics are the curves obtained as a result of the intersection of one of the surfaces of a cone with another plane surface. These sections differ in their properties. A cone has two identical shaped parts that we usually refer to as the cone itself. These parts are called the nappes. A nappe resembles the shape of a birthday hat. Depending upon the point of intersection, the conic section can be of different types. These sections are of three basic types: parabola, hyperbola, and ellipse. A circle is another example of a special case of conic section types. In ancient times it was considered as the 4th type of it.

Parameters or Features of Conic Sections

Conic sections have three primary features, i.e. focus, directrix, and eccentricity. The different shapes of the three types of conic sections rely on these three parameters. Thus, to understand the types, we must first understand these parameters.

• Focus: The point of the creation of a conic section is the focus of it. Different types of the conic section have different focal points. For instance, a parabola has single foci, whereas the hyperbola and ellipses both have two.
• Directrix: A line that defines these conic sections is the directrix. These lines are always perpendicular (90 degrees) to the cone’s axis. The ratio of foci to directrix determines the point on a conic section. Like the focus, a parabola has a single directrix, and the hyperbola and ellipse have two each. Special cases, such as circles, do not have a directrix.
• Eccentricity: It is the ratio of the distance of a point from the focus and directrix. They help in the determination of the shape. Eccentricity is always a positive real number. The value of eccentricity of a circle is 0. The value for the ellipse is either equal to zero or less than 1. Parabolas have an eccentricity of 1, whereas hyperbolas have an eccentricity greater than 1.

Types of Conic Sections

As discussed above, there are three types of conic sections based on the point of intersection of the plane with the curve. These are as follows:

• Parabola: The plain of a parabola is always parallel to the generating line. A parabola refers to points whose distance from the focus is equal to the distance from the directrix. At the midpoint of these two i.e., focus and directrix is the vertex of the parabola. A quadratic function always shows this type of graph, i.e., a parabolic graph.
• Ellipse: A conic section is an ellipse if the plane intersects a nappe at any angle to the axis but 90 degrees. In an ellipse, the sum of distances from two foci remains constant. Thus, in an ellipse, there will be two foci and directrices.

Assume an ellipse with centres as h, k; the length of major axes as 2a and minor axes as 2b. The formula derived for it will be as follows:

(x−h)2/a2 + (y−k)2/b2 = 1

• Hyperbola: A conic section is a hyperbola only if the plane is parallel to the y-axis (i.e. the axis of revolution). In a hyperbola, the difference between two foci remains constant. Like the ellipses, they also have two foci and directrices. Assume the same conditions as in the ellipse. Then the derived formula for the hyperbola will be:

(x−h)2/a2 – (y−k)2/b2 = 1

•  The circle: As already said, it is a special case and a type of ellipse, where the plane is parallel to the base of the cone. The foci of the circle are the centre of that circle. It has no directrix and 0 value for eccentricity. The equation for the circle with radius r is as follows:

r2= (x−h)2 + (y−k)2 

Solved Question on Conic Sections

Q1. In an ellipse, the foci lie at coordinates (3, 0). The vertex and the centre are (4, 0) and (0, 0), respectively. What will be the equation for this ellipse?

Solution: According to the question, we assume that:

c = 3 and a = 4.

We use, b2 = a2 – c2

b2= 16 – 9

= 7

Substituting this in the ellipse conic section’s equation, we get:

x2/a2 + y2/b2 = 1

x2/16 + y2/7 = 1

Answer: The equation for the ellipse in the given case shall be x2/16 + y2/7 =1.

Conclusion

Curves obtained by intersecting one of the surfaces of a cone with another plane surface are known as conic sections. A cone has two symmetrical parts that we usually refer to as the cone itself. These parts are called the nappes. These sections are of three basic types: parabola, hyperbola, and ellipse. A circle is also an example of a type of conic section. Focus, directrix, and eccentricity are the three primary features of conic sections. The different shapes of the three types of conic sections rely on these three parameters. Explore the article for study material notes on the conic sections.