M4. Conic Section:

Conic sections are the geometric curve formed when a cone is cutting by a plane. 

When double-napped cone is slicing by a plane, then different types of conic sections are obtained as shown in figure. (Click here for more detail.)

The Conic section are divided into 4 - major categories as follows:

1) Circle

If a plane intersects a cone perpendicular to the axis, then the section is a circle, as shown in figure above.

2) Ellipse

If a plane intersects a cone at a given angle with the axis greater than the semi - vertical angle. then the section is an ellipse.

3) Parabola

If an intersection plane, not passing through vertex, is parallel to the generator of the cone, then the section is a parabola.

4) Hyperbola

If a plane intersects the double right cone such that the angle between the axis and the plane be less than the semi-vertical angle, then the section is a hyperbola.

» Generally, Conic Section is defined as:

The locus of a point which moves in a plane in such a way that the ratio of its distance from a fixed point to its distance from a fixed straight line is constant is called a conic section.

Where, the straight line is called directrix. The fixed point is called the focus. The constant ratio is an eccentricity (denoted by $e$). 

$eccentricity\;(e) = \frac{distance\;from\;focus\;to\;conic}{distance\;from\;conic\;to\;directrix}$

The straight line passing through the focus and perpendicular to the directrix is called the axis. The intersection of the curve and the axis is called the vertex.

A Conic section are classified by its eccentricity as follows:

If eccentricity $(e) = 0$; the conic is a Circle. 

If eccentricity $(e) = 1$; the conic is a Parabola. 

If eccentricity $(e) < 1$; the conic is an Ellipse. 

If eccentricity $(e) > 1$; the conic is a Hyperbola.

 ..........................************............************ ..........................

1. Parabola:

Parabola has vertical axis

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

$(y-k)^2 = 4a(x-h)$; where $a≠0$

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

$y^2 = 4ax $ ............ (i)

                                                                         Where $a≠0$; Parabola has a horizontal axis.

                                                                            Similarly,

                                                                           $x^2 = -4ay $ .............. (ii)

                                                                           Where $a≠0$; Parabola has a vertical axis.

» The important results of a parabola are tabulated below:

Equation of Parabola	Vertex	Focus	Equation of directrix	Axis	Length of latus Rectum
$y^2 = 4ax$	$(0,0)$	$(a,0)$	$x=a$	$y=0$	$4a$
$x^2 = -4ay$	$(0,0)$	$(0,a)$	$y=-a$	$x=0$	$4a$
$(y-k)^2 = 4a(x-h)$	$(h,k)$	$(h+a,k)$	$x= h-a$	$y=k$ parallel to $x-axis$	$4a$
$(x-h)^2 = 4a(y-k)$	$(h,k)$	$(h,k+a)$	$y=k-a$	$x=h$ parallel to $y-axis$	$4a$

2. Ellipse:

Ellipse has major axis $'a'$ as horizontal

Let an ellipse in the rectangular coordinate plane with the center at the origin $(0,0)$ and the major axis along the x-axis as shown in figure.

$\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$ .......... (i)

Where $'a'$ is major axis as horizontal and $'b'$ is minor axis as vertical.

Similarly,

                 $\frac{x^2}{b^2}+\frac{y^2}{a^2} = 1$ .......... (ii)

Where $'b'$ is a major axis as vertical and $'a'$ is minor axis as horizontal.

If center are not at origin, that is the center will be shifted to the point $(h,k)$ then the equation of ellipse becomes;

                 $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ .......... (iii)

Where $'a'$ is major axis as horizontal and $'b'$ is minor axis as vertical.

Similarly,

                 $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$ .......... (iv)

Where $'b'$ is a major axis as vertical and $'a'$ is minor axis as horizontal.

» The important results of a Ellipse are tabulated below:

3. Hyperbola:

Hyperbola opens right and opens left

The standard form of an equation of a hyperbola with a horizontal transverse axis and center at the origin $(0,0)$ is given by: 

$\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1$ .......... (i)

Where $a≠0$ and $b≠0$.

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

The standard equation of an equation of a hyperbola with a vertical transverse axis and center at the origin $(0,0)$ is given by:

                $\frac{y^2}{a^2}-\frac{x^2}{b^2} = 1$ .......... (ii)

Where $a≠0$ and $b≠0$.

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

» The important results of a Hyperbola are tabulated below:

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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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CS2. Conic Section (Part 2):

In previous lecture we already mentioned that 'How developed the Conic Section is'?

There are several way to approach the study of conic. But now we are trying to introduce basically what conic section is? What are its types?

Conic sections are commonly studied topic of geometry. They play an important role in both mathematics & physical process in nature.

Fig1: Conic Section

Conic sections are the geometric curve formed when a cone is cutting by a plane. When double-napped cone is slicing by a plane, then different types of conic sections are obtained. A doubled-napped cone is 3-dimensional but formations of conic sections are 2-dimensional curves or plane curves. So the desirable definition of conic section avoid the notion of a cone. 

The conic sections are divided into 4-major categories as follows:

1) Circle; 

                                          2) Ellipse; 

                                          3) Parabola; 

                                          4) Hyperbola.

1. Circle: 

When a cone is cutting by the plane perpendicular to its conic axis, circle is formed.

2. Ellipse: 

When 3-dimensional cone is cutting by a plane which is not perpendicular to its conic axis  or not parallel to the generator, but cutting a cone in a closed curve, ellipse is formed.

3. Parabola:  

If 3-dimensional cone is cutting by a plane is parallel to one & only one generator then the conic is called a parabola.

  
4. Hyperbola: 

If 3-dimensional cone is cutting by a plane is parallel with two generator of a cone, which intersects both napped of a cone. This types of conic is called a hyperbola.

Fig2: Conic Section

Note that: The intersecting plane does not passing through the vertex of the cone. When plane does passes through the vertex the degenerating conic are formed. Degenerating conics: which include a point, a line, pair of parallel line & pair of intersecting lines. The figures are shown as follows:

Fig3: Degenerated Conic

If you want more demonstration in this topic, click here: Interactive 3-D conic Graph.

Definition of conic section:

Fig4: Conic Geometrically

A conic section is the locus of a point, which moves in a plane such that its distance from a fixed point is in a constant ratio to its distance from a fixed straight line.
Where,

•  The collection of points is called locus.

•  The fixed point is called focus.

•  The fixed line is called the directrix.

•  The constant ratio is called the eccentricity. It is denoted by 'e'.

 In the figure, '$P$' is a point & '$Q$' is the foot of a line from '$P$' perpendicular to the directrix. Where,

$\frac{FP}{PQ} = Constant = e \Rightarrow FP = ePQ$ 

The term eccentricity is defined as the ratio of focal distance to directrix distance of the conic section. It is a measurement of how much a conic deviated from being conic circular.

$eccentricity\;(e) = \frac{distance\;from\;focus\; to\;conic}{distance\;from\;conic\;to\;directrix}$  

The conic sections are classified by its eccentricity as follows:

•  If $e = 0$; the conic is a circle.

•  If $e = 1$; the conic is a parabola.

•  If $e < 1$; the conic is an ellipse.

•  If $e > 1$; the conic is a hyperbola.

General equation of conic section:

Let us move from geometric conic to algebraic conic:

The equation of the conic is represented by the general equation of 2^nd degree.

    

Where $A$, $B$, $C$, $D$, $E$, $F$ are constant. If we change the value of these constants, the shape of the corresponding conic will also change. So we have to focus on these differences in the algebraic equations.

The quantity $B^2 - 4AC = 0$ is the discriminant of above 2^nd degree equation . If constant $A$, $B$, $C$ are not equal to zero, it can help to determine the shape of the conic.

A conic sections can be classified by its discriminant as follows:

• If  $C = A$ and $B = 0$, the conic is circle.

• If  $B^2 - 4AC = 0$, the equation represents a parabola, a line or else has no graph.

• If $B^2 - 4AC < 0$, the equation represents an ellipse, a point or else has no graph.

• If $B^2 - 4AC > 0$, the equation represents a hyperbola, a pair of intersecting lines or else has no graph.

We will be discuss about 'Conic Sections & its Importance' later on.

CS1. Conic Section (Part 1):

>> Mathematicians have a habit of studying, just for the fun of it, things that seem utterly useless; then centuries later their studies turn out to have enormous scientific value....!

In this section, we have to discuss about 'Conic Sections' especially trying to simply introduce - How developed the 'Conic Sections' concept's is?

How developed the 'Conic Sections' Concepts:

Conic sections is among the oldest curve, and is an old mathematics topics studied symmetrically & thoroughly.

Fig: Conic Section

Firstly, the idea of the 'Conic Sections' was conceived in an attempt to solve the three famous construction problems of: trisecting the angle, doubling the cube & squaring the circle.

Menaechmus (380-320 BC) was the ancient Greek mathematician, who 1^st construction involving a 'Cone' in mind to solved the problem of 'doubling the cube'. He developed the properties of conic section by using the concept 'cutting of a cone'. But he did not use the term 'Parabola' & 'Hyperbola' to deal with such types of curve. Instead of these, he called 'section of a right-angled cone' for parabola & 'section of an obtuse-angled cone' for hyperbola. [Remind that: these term 'parabola' & 'hyperbola' are named as by Apollonius].

After Menaechmus, Aristaeus & Euclid formalized & expanded upon the 'Conics' through their written books. They wrote various types of books (of solid loci) connected with the 'Conics'.

Then came the greatest Archimedes, who used the elementary theory of conic sections to developed important concepts about parabolas.

Apollonius (262-190 BC) was the greatest ancient Greek mathematician, who consolidated & extended previous results of conics into a monograph conic sections, consisting of eight books with 487 propositions. He also completed the four books of 'Euclid's Conics'. He was the first, to base the theory of all three conics (ellipse, parabola & hyperbola) on sections of one circular cone, right or oblique. He also named the conic sections as: Ellipse, Parabola & Hyperbola.

Fig: Kepler's planets orbit

After Apollonius, the Conic Sections theory disappeared for over 1000 years; No any important scientific applications were found until 17^th century. After 17^th century, Kepler & Galileo helps to develop the 'Conic Sections' by using for scientific approach. Kepler derived his first law of planetary motion(1609). He discovered that Planets move in elliptical path with the Sun at one focus. Galileo proved that projectiles travel in parabolas - in his Dialogues of the Two New Sciences(1638). They used to powerful modeling tools (i.e. Conics) for explaining the physical laws of the universe.

In 1639, the French engineer Girard Desargues initiated the study of those properties of conics that are invariant under projections (See: Projective geometry). In 18^th century, architects 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.
Likewise, now-days new application for conic sections continue to be found. (See more in: Conic Sections & its Importance)

After then everybody knows that - How powerful & how useful these 'Conic Sections theory' is?

In the next note, we will discuss about: What 'Conic Sections' is? & their types....