Generation of Conics

Conic sections are the curves obtained when a plane intersects the surface of a hollow double cone. Conic sections are the set of points whose coordinates satisfy a quadratic equation in two variables.

Like, x2 + y2 -2x – 3 = 0

(y-2)2 = 49x

9x2 +4y2 -36 = 0

x2-y2 = 1

There are different conic sections in mathematics based on the angle formed between the plane and the intersection of the right circular cone with it. Conic sections are divided into

1. Degenerate curves – for example, point, line, a pair of interesting lines

2. Non-degenerate curves – for example, circle, ellipse, parabola, hyperbola

The cone is not necessarily a right circular cone. There are many conic sections on the basis of the angle of cutting. 

Double-napped right circular hollow cone

Let,

 l= a fixed vertical line 

m = another line intersecting a fixed vertical line at a fixed point V 

α = angle inclined between m and l 

Try to rotate the line m around line I, 

Thus, the angle α remains constant

As a result, a double-napped right circular hollow cone is generated which was later referred to as cone.

This cone can extend indefinitely in both directions.

Point Vertex = point of intersection of 2 cones

line l = Axis of cone

rotating line m = generator of the cone

Vertex separates two parts of the cone into two parts called nappes.

On taking the intersection of a plane with a cone, you will obtain a section called the conic section.

β is the angle formed after the intersection of the plane with the vertical axis of the cone. When the plane cuts nappe of the cone, different types of conic sections can be formed, depending upon β:

1. β = 90° = circle

2. α < β < 90° = ellipse

3. β = α = parabola

4. 0 ≤ β < α = hyperbola

Conic section Circle

A circle is the set of all points in a plane situated at a fixed distance from a fixed point in the plane.

Fixed point = center of the circle

Distance from center to any point on the circle = radius of the circle

Conic section circle formula:

(x – h)2 + (y – k)2 = r 2

Where, r = radius of the circle; center of the circle = (h,k)

General equation:

x2 + y2 + 2gx + 2fy + c = 0

Where, g, f, and c are constants

             Center of this circle = (-g, -f)

             Radius of the circle = √g2 + f2 – c

Conic section Parabola

A parabola is a conic section. It is a section of a right circular cone parallel to one side (a generating line) of the cone.

A parabola can be defined as the set of all points in a plane that are an equal distance away from a given point(focus) and a given line (directrix).

Fixed point of parabola = Focus (F)

Fixed line (l) of parabola = Directrix of parabola

Standard equation of parabola

                             (c)                                                                           (d)

Conic section Ellipse

The locus of all those locations in a plane whose sum of distances from two fixed points in the plane is constant is called an ellipse. The foci refers to the fixed points.

It is a conic section formed by the intersection of a right circular cone by a plane that cuts the axis and the surface of a cone.

Eccentricity of the ellipse is less than one.

Fixed point = focus

Fixed line = directrix

Constant ratio (e) = eccentricity of the ellipse

In an ellipse, centricity is between zero to one, i.e. 0<e<1

Two standard forms of an ellipse are:

1. Major axis along the x-axis and minor axis along the y-axis

x2/a2 + y2/b2 = 1

2. Major axis along the y-axis and minor axis along the x-axis

x2/b2 + y2/a2 = 1

Conic section Hyperbola

Hyperbola is made of two pieces that are mirror images of each other. A section on Conics The set of all points in a plane whose distances from two fixed points are equal is known as a hyperbola.

Fixed point = focus

Fixed line = directrix

Constant ratio = eccentricity of hyperbola (e)

Components of hyperbola:

1. Major axis – The length of the major axis is 2a in the hyperbola.

2. Minor axis – The length of the minor axis is 2b in the hyperbola.

3. Foci – The hyperbola has two foci,  (ae, 0) and (-ae, 0) .

4. Vertices of hyperbola – It is the point where the hyperbola intersects the major axis.

5. Center of hyperbola – It is the midpoint of the line that joins two foci of the hyperbola.

Two standard forms of hyperbola are

• Transverse axis along the x-axis and conjugate axis along with the y-axis

x2/a2 – y2/b2= 1

• Transverse axis along the y-axis and conjugate axis along the x-axis

y2/a2 – x2/b2= 1

In hyperbola, eccentricity (e) is more than 1, i.e., e>1

 Conclusion

Conic sections are two -dimensional planes that cut through from a cone. Different types of conic sections are circle, parabola, ellipse, and hyperbola. Standard forms of different conic sections are

1. Circle: x2+y2=a2

2. Parabola: y2=4ax when a>0

3. Ellipse: x2/a2 + y2/b2 = 1

4. Hyperbola: x2/a2 – y2/b2 = 1