CS3. Conic Section & its Importance (Part 3):

In previous lecture we discussed 'What Conic Section is'? & 'It's Development'. Now in this section we will discuss the remaining part of conic section - 'Conic Section & its Importance':

We already know that, Conic Section is the curve intersection by a plane with a doubled napped 3-dimensional cone. By changing the angle & location of the intersection, 4-different types of conic sections are formed as follows:

1. Circle

2. Ellipse

3. Parabola

4. Hyperbola

1. Circle: 

This is one of the basic types of conic section is called circle.

Fig: Circle

A circle is defined as the set of all points in a plane that are equidistant from a fixed point called the center.

The fixed distance from the center is called the radius.It is denoted by '$r$'; where $r > 0$.

Standard equation of Circle:

Let $(x, y)$ be any arbitrary points on the circle. And a circle is centered at the points $(h, k)$ & having radius$(r)$.

Then by definition: the distance between $(h, k)$ & $(x, y)$ must be radius $(r)$ is:

$\left ( x-h \right )^2 + \left ( y-k \right )^2 = r^2$

If a circle is centered at the origin, i.e. $(h, k) = (0, 0)$ then the equation of a circle is:

$x^2 - y^2 = r^2$

Importance:

• The circle symbol meaning is universal, sacred & divine. It represents the infinite nature of energy & the inclusive of the universe.

• The circle is one of the most common & important shapes in our daily life. One of the greatest inventions of the circle is wheel. Which are generally used in vehicles to transportation system in our everyday life.

• Helicopter & Plane are also could not fly, as the blades that it into the air spin round in circle.

• The circle is one of the complex shapes. Indeed, very difficult for man to create the circular shape but nature manages to do it perfectly. The movement of planets, natural cycles, natural shapes - like as: center of flower, eyes, human head & many other things are circular in shape, which we see them in our everyday life.

• There are numerous physical circular equipment's, which are used in our everyday life. Some of them are: wheels, CDs, DVDs, plates etc.

• Circles are also used to consideration the shape of number to represent zero (0).

2. Ellipse:

Another basic types of conic section is called ellipse. It is closed 2-dimensional curve look like a flattened circle. It is differ from circle, does not have constant radius.

Fig: Ellipse

An ellipse is the set of all points $(x, y)$ in a plane, such that sum of the distance between $(x, y)$ two distinct fixed points is constant. This two fixed points are called foci of an ellipse.

Standard equation of Ellipse:
The standard equation of an ellipse with its center at the point $(h, k)$; having length of major axis '$2a$' & length of minor axis is '$2b$' (where '$a$' & '$b$' are positive real number) is:

 $\mathbf{\frac{\left ( x - h \right )^2}{a^2} + \frac{\left ( y - k \right )^2}{b^2} = 1}$; Where '$a$' is major axis as horizontal & '$b$' is minor axis as vertical.

  Or,

$\mathbf{\frac{\left ( x - h \right )^2}{b^2} + \frac{\left ( y - k \right )^2}{a^2} = 1}$; Where '$b$' is major axis as vertical & '$a$' is minor axis as horizontal.

If an ellipse with the center at the origin, having length of major axis '$2a$' & length of minor axis '$2b$' respectively (where $0 < b < a$) is:

$\mathbf{\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1}$; Where '$a$' is major axis as horizontal & '$b$' is minor axis as vertical.

Or,

$\mathbf{\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1}$; Where '$b$' is major axis as  vertical & '$a$' is minor axis as horizontal.

Note: The equation of ellipse reduces to a circle in the special case in which '$a = b$' is:

$\mathbf{x^2 + y^2 = a^2}$

Importance:

• The ellipse is one of the most important curves in physical science. In astronomy, the orbit of each planet is an ellipse with one focus at the center of the sun.

• In room where ceilings are elliptical, a sound made at one focus can be heard very clear at another focus. Which is proved by the architects in 18^th century, created a fad for Whispering galleries - such as in the St. Paul's Cathedral in London - in which a Whisper at one focus of an ellipsoid can be heard at the another focus, but cannot be heard at many places in between.

• On a far smaller scale, the electrons of an atom move in an approximately elliptical orbit with the nucleus at one focus.

• An ellipse is also used in Lithotripsy, a medical procedure for treating kidney stones. The patient is placed in an elliptical tank of water a kidney stone at one focus. High energy shock waves generated at another focus are concentrated on the stone & then pulverizing it.

• An ellipse has another important property that is used in the reflection of light & sound waves. Any light or signal that starts at one focus will be reflected to another focus of an ellipse.

3. Parabola:

Fig: Parabola

A parabola is the set of all points $(x, y)$ in a plane that are equidistant from a fixed line & fixed points. The fixed points is called focus & the fixed line is called directrix. The distance from fixed points is equal to its distance from a fixed line as shown in figure:

Standard equation of Parabola:

Let $P = (x, y)$ be a point on a parabola. The standard equation of a parabola with its vertex $(h, k)$ & focus is at $(h + p, k)$ is given by:

$\mathbf{(y-k)^2 = 4p(x-h)}$; Where $p\neq 0$.

The parabola is parallel to x-axis (i.e. parabola has a horizontal axis). If $P >0$; the parabola opens to the right. And if $P <0$, the parabola opens to the left.

Or,

The standard equation of parabola with its vertex $(h, k)$ & focus is at $(h, k + p)$ is given by:

 $\mathbf{(x-h)^2 = 4p(y-k)}$; Where $p\neq 0$

The parabola is parallel to y-axis (i.e. parabola has a vertical axis). If $P > 0$, the parabola opens to the upward. And if $P < 0$, the parabola opens to the downward.

Note: If a parabola is centered at the origin $(h, k) = (0, 0)$ & focus is at $(p, 0)$, then the equation of the parabola is:

$\mathbf{y^2 = 4px}$, Where $p\neq 0$; Parabola has a horizontal axis.

 Or,

$\mathbf{x^2 = 4py}$, Where $p\neq 0$;Parabola has a vertical axis.

Importance:

• Everything on the planets earth is subjected to gravity. When an object is thrown or forced into the air, eventually gravity grabs it & drags down on the ground. The path of the object is Parabolic.

• A parabolic mirror has the property that incoming rays  of light are reflected in a path parallel to the symmetry axis. This property is an essential feature of satellite dishes, radio telescopes & reflecting telescopes including liquid mirror telescopes.

• The cables holding up the Golden Gate Bridge form parabolas. Because they helps to carry the forces acting on the bridge to the top of the towers.

• Many of the older space missions (spacecraft) followed a parabolic path. If we launched the spacecraft directly into the atmosphere, that would cause it to burn up.

• Parabolic antennas catch & focus radio waves from distant galaxies. The nose of an airplane has a parabolic shape, that helps to reduce wind resistance. The parabolic-shaped ears of some animals, that helps them to catch sounds properly.

4. Hyperbola:

Fig: Hyperbola

Hyperbola is the 2-dimensional curve with two mirror image.

A hyperbola is the set of all points $(x, y)$ in a plane, the difference of whose distances from two distinct fixed points is constant. These two fixed points is called foci. The graph of the hyperbola has two disconnected parts - called branches. A hyperbola has two vertices that lie on an axis of symmetry called the transverse axis. The transverse axis is either horizontal or vertical.

Standard equation of Hyperbola:

The standard form of an equation of a hyperbola with a with a horizontal transverse axis & center at the origin is given by:

$\mathbf{\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1}$; Where $a\neq 0$ & $b\neq 0$

The x-term is positive. The branches of the hyperbola opens left & opens right.

The standard form of an equation of a hyperbola with a vertical transverse axis & center at the origin is given by:

$\mathbf{\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1}$; Where $a\neq 0$ & $b\neq 0$

The y-term is positive. The branches of the hyperbola opens up & opens down.

Note: Here, the definition of hyperbola is similar to that of an ellipse. The distinction is that: for an ellipse is the sum of the distances between the foci & a point on the ellipse is constant, where as for the hyperbola is the difference of the distances between the foci & a point on the hyperbola is constant.

Importance:

• There are many natural phenomena that involve hyperbolas. For example: Sonic booms are created when an object exceeds the speed of sound in air. The shock wave of a sonic boom takes the shape of a cone & when it interacts the ground, it takes the shape of a hyperbola. Every point on the curve is hit at the same time. So everyone on the ground will hear the sound at the same time.

• Sound waves travel in hyperbolic paths.

• Radio system's signals employ hyperbolic functions to optimize the area covered by the signals from a station.

• Hyperbolic gears are used in many machines & in the industry.

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