Conics in Real Life

Introduction

• Conics or conic sections were studied by Greek mathematicians, with Apollonius of Pergo’s work on their properties around 200 B.C.

  
  

• Conics sections are planes, cut at varied angles from a cone. The shapes vary according to the angle at which it is cut from the cone.

• As they are cut from cones, they are called Conies. Further, they have some common properties as they all belong to cones. These curved sections are related to.

  
  

Also read:

• Formulas of Coordinate Geometry
• Algebra in Real Life
• Probability in Real Life
• Parallel and Perpendicular Lines in Real Life
• Algebra in Real Life
• Geometry in Real Life

What is Conic Section?

• Conic section is a curve obtained by the intersection of the surface of a cone with a plane.

• In Analytical Geometry, a conic is defined as a plane algebraic curve of degree 2. That is, it consists of a set of points which satisfy a quadratic equation in two variables. This quadratic equation may be written in matrix form. By this, some geometric properties can be studied as algebraic conditions.

• Thus, by cutting and taking different slices(planes) at different angles to the edge of a cone, we can create a circle, an ellipse, a parabola, or a hyperbola, as given below

  
  

• The circle is a type of ellipse, the other sections are non-circular. So, the circle is of fourth type.

Focus, Directrix and Eccentricity

The curve is also defined by using a point(focus) and a straight line (Directrix).

If we measure and let

a – the perpendicular distance from the focus to a point P on the curve,

and b – the distance from the directrix to the point P,

then a: b will always be constant.

\(a: b < 1\) for ellipse

\(a: b= 1\) for parabola as \(a= b\)

and \(a: b> 1\) for hyperbola.

Eccentricity: The above ratio a: b is the eccentricity.

Thus, any conic section has all the points on it such that the distance between the points to the focus is equal to the eccentricity times that of the directrix.

Thus, if eccentricity \(<1\), it is an ellipse.

if eccentricity \(=1\), it is a parabola.

and if eccentricity \(=1\), it is a hyperbola.

For a circle, eccentricity is zero. With higher eccentricity, the conic is less curved.

Latus Rectum

The line parallel to the directrix and passing through the focus is Latus Rectum.

Source: i2.wp.com/c1.staticflickr.com

Length of Latus Rectum = 4 times the focal length

Length \(=\frac{2b^2}{a}\) where \(a =\frac{1}{2}\) the major diameter

and \(b =\frac{1}{2}\) the minor diameter.

= the diameter of the circle.

Ellipse has a focus and directrix on each side i.e., a pair of them.

Equations

General equation for all conics is with cartesian coordinates x and y and has \(x^2\) and \(y^2\) as

the section is curved. Further, x, y, x y and factors for these and a constant is involved.

Thus, the general equation for a conic is

\[Ax^2 + B x y + C y^2+ D x + E y + F = 0\]

Using this equation, following equations are obtained:

For ellipse, \(x^2a^2+y^2b^2=1\)

For hyperbola, \(x^2a^2-y^2b^2=1\)

For circle, \(x^2a^2+y^2a^2=1\) (as radius is a)

Parabola in Real Life

• Parabola is obtained by slicing a cone parallel to the edge of the cone. It is of U – shape as a stretched geometric plane. This formula is \(y =x^2\) on the x – y axis.

• Mathematician Menaechmus derived this formula.

  
  

• Parabola is found in nature and in works of man.

• Water from a fountain takes a path of parabola to fall on the earth.

  
  

• A ball thrown high, follows a parabolic path.

• A roller coaster takes the path of rise and fall of a parabolic track of the sea.

• An architectural structure built and named The Parabola in London in 1962 has a copper roof with parabolic and hyperbolic linings.

  
  

• The Golden Gate Bridge in San Francisco in California is famous with parabolic spans on both sides.

  
  

• In light houses, parabolic bulbs are provided to have a good focus of beam to be seen from distance by mariners.

  

• Automobile headlights are also with parabola type.

  
  

• The stretched arc of a rocket launch is parabolic.

  
  

• The satellite dish is a parabolic structure facilitating focus and reflection of radio waves.