Conic Sections

A conic is an degenarate curves coming out as a result of intersection of plane with one and two Nappies. For the plane perpendicular to the axis of the cone a circle is generated. On a plane a parabola or ellipse is created. On double planes of a cone a hyperbola is created.

Conic

History of Conic Sections

A conic section is a curve of the intersection of a plane with a right circular cone. The Greek Mathematician Appollonius studied conic sections in about 240 B.C in terms of geometry, by using this concept. In the geometry of conic sections, a cone is regarded as having two nappes extending indefinitely in both directions. A generator of the cone is a line lying in the cone.

The curve of intersection of a plane with a cone depends on the inclination of the axis of the cone to the cutting plane.
                                                                                     Appollonius
       
If the cutting plane is parallel to one and only one generator then the conic is a parabola. If the cutting plane is parallel to no generator, then the conic is an ellipse.
When the cutting plane is parallel to two generators, then it intersects both nappes of the cone and the conic is called a hyperbola.
In geometry, conic section is the locus of a point which moves so that its distance from a fixed point bears a constant ratio to its distance from a fixed line.

Conic Section Definition

The locus of a point which moves from a plane such that the ratio of its mean distance from a point to its perpendicular distance from a fixed straight line is always constant, is known as a conic section or a conic.

Consider the following figure.

Conic Section
In the figure,$\frac{SP}{PM}$ = Constant=e $\Rightarrow$ SP = e PM
The fixed point(S, S’)  is called the focus of the conic and this fixed line is called the directrix of the conic. Also this constant ratio is called the eccentricity of the conic and is denoted by e.

Identifying Conic Sections

If e = 1, the conic is called Parabola.
If e < 1, the conic is called Ellipse.
If e > 1, the conic is called Hyperbola.
If  e = 0, the conic is called circle.
If  e = ∞ , the conic is called pair of straight lines.
Identifying conic sections:  The above equation of conic sections can be recognized by the condition given in the tabular form. For this, we first need to find the discriminant of the equation which is,
∆ = abc + 2fgh – af2 – bg2 – ch2

Conic Sections Equations

Conic sections equations: The equation of conics is represented by the general equation of the second degree
ax2 + 2hxy + by2 + 2gx + 2fy + c =0 -------------------------- ( 1 )

Case 1: When ∆ = 0.

In this equation (1) represents the Degenerate conic whose nature is given in the following table.
     Condition 
          Nature of Conic
 ∆ = 0 and ab – h2 = 0   A pair of straight lines or empty set.
 ∆ = 0 and ab – h2 ≠ 0  A pair of intersecting straight lines.
 ∆ = 0 and ab – h2 < 0  Real or imaginary pair of straight lines
 ∆ = 0 and ab – h2 > 0  Point

Conic Sections Formulas

Case 2: When  ∆ ≠ 0.

In this case equation (1) represents the Non-degenerate conic whose nature is given in the following table:
        Condition
     Nature of Conic               
 ∆ ≠ 0, h=0, a=b  A circle
 ∆ ≠ 0, ab – h2 = 0  A parabola
 ∆ ≠ 0,  ab – h2 > 0  An Ellipse or empty set
 ∆ ≠ 0,  ab – h2 < 0  A hyperbola
 ∆ ≠ 0,  ab – h2 < 0 and a+b=0  A rectangular hyperbola

Conic Sections Circles

Conic Sections Circles: The second degree equation ax2 + 2hxy + by2 + 2gx + 2fy + c =0, represents a circle if,
                       ∆  = abc + 2fgh – af2 – bg2 – ch2 ≠ 0 , h = 0 and a = b


Conic Sections Circles
The above equation will become,
ax2 + ay2 + 2gx + 2fy + c =0, which represents the equation of  a circle whose center is ($\frac{-g}{a},\frac{-f}{a}$) and
Radius r = $\sqrt{(\frac{-g}{a})^{2}+(\frac{-f}{a})^{2}-C}$

Example : What does the conic,  x2 + y2 – 6x + 4y – 8 = 0, represent?

Solution: We have the equation, x2+ y2– 6x + 4y – 8 = 0.
Comparing this with the second degree equation, ax2 + 2hxy + by2 + 2gx + 2fy + c =0,
we observe that a = b = 1, and h = 0.
From the table it is clear that this represents the equation of circle.
Let us evaluate, ∆  = abc + 2fgh – af2 – bg2 – ch2
                           = -8 + 0 -4  - 9 – 0
                           = -21     ≠ 0.
Hence the above system of equation represent the equation of a circle
whose centre is ( -g, -f) = (3, -2) and  radius = $\sqrt{9+4-(-8)}$
                                                                = √21 units    

Conic Sections Ellipse

Conic sections Ellipse: The second degree equation ax2 + 2hxy + by2 + 2gx + 2fy + c =0, represents an ellipse if,
                                   ∆  = abc + 2fgh – af2 – bg2 – ch2 ≠ 0  and ab – h2 > 0.

Example : What conic does 13x2 – 18 xy + 37y2+ 2x + 14y -2 = 0 represent?

Solution: We have, 13x2 – 18 xy + 37y2+ 2x + 14y -2 = 0
comparing this with the second degree equation, ax2 + 2hxy + by2 + 2gx + 2fy + c =0
we get, a= 13, b= 37,c=-2
2h = -18 => h = -9
2g = 2 =>   g = 1
2f = 14 =>  f= 7
Therefore, ∆  = abc + 2fgh – af2 – bg2 – ch2 = (13)(37)(-2) +2(7)(1)(-9) – (13)(7)2 – 37(1)2- (-2)(-9)2
                   = -962 -126 -637 -37 + 162
                   = - 1600 ≠ 0
and also ab – h2 = (13)(37) – (-9)2 = 481 – 81 = 400 > 0
Hence the conditions are satisfied as per the above table for an ellipse.
Therefore the given equation represents equation of an ellipse.

The General Equation:
The general, the equation of an ellipse is, $\frac{(x-h)^{2}}{a^{2}}$ + $\frac{(y-k)^{2}}{b^{2}}$ = 1, where a > b or a < b.
If a= b, the above equation becomes the equation of a circle.
Let us consider the equation, $\frac{(x-h)^{2}}{a^{2}}$ + $\frac{(y-k)^{2}}{b^{2}}$ = 1,where a>b.
The co-ordinates of its center is ( h, k)
The Major axis is x-axis and Minor axis is y-axis.
Length of the major axis is 2a and length of the minor axis is 2b.
The co-ordinates of the vertices are ( ± a, 0 ), ( 0, ± b)
The co-ordinates of foci are ( (ae+h, k) and (-ae+h, k)
The equation of directrices are x = $\frac{a}{e}$ + h and x = - $\frac{a}{e}$ + h
The eccentricity is calculated using the relation b2 = a2 ( 1 – e2 )
Length of the latus rectum is $\frac{2b^{2}}{a}$

Graphing Conic Sections

Example: Graph the conic section whose equation is given by, 25 x2 + 16 y2  = 1600

Solution: We have, 16 x2 + 25 y2  = 1600
$\Rightarrow$ $\frac{16x^{2}+25y^{2}}{1600}$ = $\frac{160}{1600}$ =1
$\Rightarrow$ $\frac{x^{2}}{100}$ + $\frac{y^{2}}{64}$ = 1
$\Rightarrow$ a2 = 100 and b2 = 64
$\Rightarrow$ a = ± 10 , and b = ± 8, since a > b, the equation represents the ellipse whose major axis is horizontal.
Since h = k = 0, The center of the ellipse is (0, 0 )
Calculation of e:
We have, b2 = a2 ( 1 – e2 )
$\Rightarrow$   64 = 100( 1 – e2 )
$\Rightarrow$   64 = 100 – 100 e2
$\Rightarrow$   100 e2 = 100 – 64 = 36
$\Rightarrow$   $e^{2}$ =$\frac{36}{100}$
$\Rightarrow$   $e$ = $\frac{6}{10}$ = $\frac{3}{5}$ = 0.6< 1
$\Rightarrow$   e = 0.6 or $\frac{3}{5}$ The co-ordinates of foci will be, ( ± ae , 0)
$\Rightarrow$   (± (10)$\frac{3}{5}$,0)
$\Rightarrow$   (± 6 , 0) 
To find the equation of directrices:
The equation of directrices are x = ± $\frac{a}{e}$ = ± $\frac{10}{0.6}$ = ± $\frac{50}{3}$
The ellipse can be drawn by marking, the vertices, center, focal points. The directrices are drawn parallel to the x-axis at a distance of $\frac{50}{-3}$ units [ which is 16.67 units ]
The graph of the Ellipse is shown below.


Graph of Ellipse

Conic Sections Practice

Question 1:Find the parametric equation of conic section, the equation is x2 + y2 + 12x + 4y - 3 = 0.
Question 2:
Find the parametric equation of conic section, the equation is x2/9 - y2/36 = 1.
Question 3:What does the conic x2+y2-8x+9y-4=0 ,represent ?
Question 4:What is the directrix for the parabola (y+1)2= -6(x-4) ?