Conics

The conic sections are curves generated by the intersection of planes with one or two nappes of a cone. (Single or a double cone). The following figure illustrates the different conic sections that can be generated.
Conic
1. Circle: A circle is generated by a plane cutting the cone perpendicular to its axis.

2. Ellipse: An ellipse is formed when a plane cuts any one nappe of the cone at an oblique angle but the plane remaining within the region of the apex and the base of the cone.

3. Parabola: A parabola is when generated in a similar way to that of the ellipse but with the plane going out of the specified region.

4. Hyperbola: A hyperbola is produced by a plane intersecting both the nappes of the cones

If the plane passes through the apex of the cone, we get straight lines or a pair of straight lines or a single point.
Geometric Definition:

A conic section may be defined more formally as the locus of a point 'P' that moves in the plane of a fixed point 'F' called the focus and a fixed line  'd' called the conic section directrix (with F not on d) such that the ratio of the distance of  P from F to its distance from d is a constant 'e' called the eccentricity.

Parameters in an Ellipse

Few parameters:

The linear eccentricity ( c ) is the distance between the center and the focus (or one of the two foci).

The latus rectum ( 2l ) is the chord parallel to the directrix and passing through the focus (or one of the two foci).

The semi-latus rectum ( l ) is half the latus rectum.
The focal parameter ( p ) is the distance from the focus (or one of the two foci) to the directrix.
The following relations hold:

  • pe = l
  • ae = c
Redefining the conics with respect to the above definition, we have

  Eccentricity 
 Conic 
    e=0
  Circle
 0  Ellipse 
  e=1  Parabola
 e>1  Hyperbola

Conic Equations

In the Cartesian coordinate system, the graph of any quadratic equation in two variables represents a conic section. This equation may be written as
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, with A, B, C not all simultaneously 0.We can then find the type of conic from the sign of B2 - 4AC

If it is less than 0, the conic is an ellipse.
If it is greater than 0, the conic is a hyperbola.
If it is equal to 0, the conic is a parabola.
If A = C and B = 0, the conic is a circle.

Example 1: Find out the type of conic section described by the equation
10x2 + 9y2 + 11xy – 32y + 48x = 20.

Solution:


Comparing the given equation with the above mentioned standard equation we get
A = 10        C = 9            E = -32
B = 11        D = 48        F = -20

B2 – 4AC = -239 (< 0). Therefore the given equation describes an ellipse.

A conic section with directrix at x = 0, focus at (p, 0), and eccentricity e > 0 has the Cartesian equation

y2 + (1 – e2)x2 – 2px + p2 = 0

We may substitute the value of p in the above equation to get the Cartesian equations of the conics. The Cartesian equations of the conics then turn out to be

Conic Formula's

    Circle 
  Ellipse   Parabola   Hyperbola 
Equation          

Variables 
    x2 + y2 = r2

r = radius of the      circle

   $\frac{x^{2}}{a^{2}}$ + $\frac{y^{2}}{b^{2}}$ = 1

     a = semi major axis
     b = semi minor axis
     c = distance from centre to focus

               y2 = 4ax


a = distance from the vertex to the focus(or directrix)

 $\frac{x^{2}}{a^{2}}$ - $\frac{y^{2}}{b^{2}}$ = 1

   a = semi major axis
   b = semi minor axis
   c = distance from centre to focus  

In the above equations, centre is taken as the origin. We may replace x by (x - h) and y by (y - k) to get a more general equation with centre at (h, k).

The parameters with respect to the above equations become

Conic section   Equation   Eccentricity (e)    Linear Eccentricity (c)   Semi-latus rectum (e)    Focal parameter (p) 
 Circle  x2 + y2 = r2    0  0  r  $\infty$
 Ellipse $\frac{x^{2}}{a^{2}}$+$\frac{y^{2}}{b^{2}}$= 1  $\frac{\sqrt{a^{2}-b^{2}}}{a}$    $\sqrt{a^{2}-b^{2}}$  $\frac{b^{2}}{a}$ $\frac{b^{2}}{\sqrt{a^{2}}-b^{2}}$
 Parabola   y2-4ax  1  a  2a  2a
 Hyperbola  $\frac{x^{2}}{a^{2}}$ - $\frac{y^{2}}{b^{2}}$  $\frac{\sqrt{a^{2}+b^{2}}}{a}$   $\sqrt{a^{2}+b^{2}}$  $\frac{b^{2}}{a}$  $\frac{b^{2}}{a^{2}+b^{2}}$

Classifying Conics

PARAMETRIC FORM OF CONIC SECTIONS

Definition: Parametric equations are method of defining a relation using parameters. A parametric equation relates two or more variables in terms of one or more independent parameters.

Each value of the parameter, when evaluated in the parametric equations, corresponds to a point along the curve of the relation.

To convert equations from parametric form into a single relation, the parameter needs to be eliminated by solving simultaneous equations.

Parametric form of circle: In the xy- Cartesian coordinate system, the circle with centre (a, b) and radius r is the set of all points (x, y) such that

(x - a)2 + (y - b)2 = r2

The equation can be written in parametric form using trigonometric ratios as

x = a + r cos t,
y = b + r sin t.

Where 't' is a parametric variable intercepted geometrically as the angle that the ray from the origin (x, y) makes with the x-axis.

If (a, b) = (0, 0) origin then the parametric equations reduces to

x = r cos t,
y = r sin t.

Where "t" is in the range of 0 to 2.

Parametric equation of Parabola: The parametric equations of a parabola are

x = 2at
y = at2

A variable point on the parabola is given by (2at, at2), for a constant 'a', and parameter 't'

Parametric equation of Ellipse: If the centre of the ellipse at a point (h, k) and the directions of the axes are parallel to the coordinate axes, then its equation is

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

The parametric form the given ellipse is given by

x = a cos t + h,
y = b sin t + k.

Parametric equation of Hyperbola: If the centre of the hyperbola at a point (h, k) and the directions of the axes are parallel to the coordinate axes, then its equation is

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

The parametric form the given hyperbola is given by

x = a sec t + h,
y = b tan t + k, for horizontal transverse axis.

x = a tan t + h,
y = b sec t + k, for vertical transverse axis.

Classifying Conics in Polar Coordinates

A conic section or a conic is the locus of a point P which moves in such a way that its distances from a fixed point S always bears a constant ratio it its distance from a fixed line, all being in the same plane.

Classifying Conic Polar Coordinates

In a polar equation of a conic, the pole is the focus of the conic, and the polar axis lies along the positive x-axis. Let ‘p’ be the distance between the focus (pole) and the directrix of a given conic. Then the polar equation for a conic takes the following form.

Formulation of Polar Coordinate:

In triangle PFB, r = $\sqrt{x^{2}+y^{2}}$ and x = r $\cos \theta$ .

Using the above identities we can derive a single equation for all the three conic sections that is Ellipse, parabola and hyperbola.

Formulation Of Polar Coordinate

In the given figure, PD = FC - FB
                                = k - r $\cos \theta$
We know eccentricity = $\frac{FP}{PD}$

$\frac{FP}{PD}$ = e

$\sqrt{x^{2}+y^{2}}$ = e.PD
                                 = e.(k-r$\cos \theta$)
r + r.e.$\cos \theta$ = e. k

r = $\frac{ek}{1 + e.\cos \theta}$

$\left\{\begin{matrix}
 0<e<1 : Ellipse & \\
 e=1 : Parabola & \\
 e>1 : Hyperbola &
 \end{matrix}\right.$

The conic section with eccentricity e > 0, focus at (0, 0) and the indicated directrix has the polar equation

  1. r = $\frac{ek}{1 + e.\cos \theta}$ ,  if the directrix is the line x = k > 0.
  2. r = $\frac{ek}{-1 + e.\cos \theta}$ , if the directrix is the line x = k < 0.
  3. r = $\frac{ek}{1 + e.\sin \theta}$ , if the directrix is the line y = k > 0.
  4. r = $\frac{ek}{-1 + e.\sin \theta}$ ,if the directrix is the line x = k < 0.

Graphing Conics

Graphing a parabola

Graphing the parabola y2 = 4ax, a > 0

The given equation y2= 4ax can be written as y = $\pm$2$\sqrt{ax}$. We observe the following

(a) Symmetry: For every positive value of x, there are two equal and opposite values of y.

(b) Region: For every negative value of x, the value of y is imaginary therefore no part of the curve lies to the left of y axis.

(c) Origin: The curve passes through the origin and the tangent at the origin is x = 0, that is y axis.

(d) Intersection with the axes: The curve meets the coordinate axes only at the origin.

(e) Portion Occupied: As x $\rightarrow$ infinity, y $\rightarrow$ infinity. Therefore the curve extends to infinity to the right of axis of y.

With the above conditions and by joining some points that satisfy the given equation we graph the parabola.

Conics Circles

Equation of circle with centre and radius:

Conic Circles

Let C be the centre of the circle and its coordinates be (h, k). Let the radius of the circle be 'a' and let P(x, y) be any point on the circumference. Then,

             CP = r

            CP2 = r2

    (x - h)2 + (y - k)2 = r2

This is the relation between coordinates of any point on the circumference and hence it the required equation of the circle having centre at (h, k) and radius a.


1. The above equation is known as the central form of the equation of a circle.

2. If the centre of the circle is at origin and radius is a, then from the above form the equation of the circle is x2 + y2 = r2

Conic Section Circles


General Equation of a circle:

The equation x2 + y2 + 2gx + 2fy + c = 0 always represents a circle whose centre is
(- g, - f) and radius

Proof:

The given equation is x2 + y2 + 2gx + 2fy + c = 0

=> (x2  + 2gx + g2 )+ (y2 + 2fy + f2) = g2 + f2 + c

=> (x + g)2 + (y + f)2 = $(\sqrt{g^{2}+f^{2}-c})^{2}$

=> [x - (- g)]2 + [y - (- f)]2 = $(\sqrt{g^{2}+f^{2}-c})^{2}$

This is of the form (x - h)2 + (y - k)2 = r2 which represents a circle having centre (h, k) and radius r.

Therefore clearly, the equation x2 + y2 = 2gx + 2fy + c = 0 always represents a circle whose centre is (- g, - f) and radius $\sqrt{g^{2}+f^{2}-c}$

1. If $\sqrt{g^{2}+f^{2}-c}$ > 0 then radius of the circle is real and hence the circle is also real.

2. If $\sqrt{g^{2}+f^{2}-c}$ = 0 then the radius of the circle is zero. Such a circle is called a point circle.

3. If $\sqrt{g^{2}+f^{2}-c}$ < 0 then the radius of the circle is imaginary but the centre is real. Such a circle is called an imaginary circle as it is not possible to draw such a circle.

4. The equation ax2 + ay2 + 2gx + 2fy + c = 0, a $\neq$ 0, also represents circle. This equation can also be written as ax2 + ay2 + $\frac{2g}{a}$x  +$\frac{2f}{a}$y  + $\frac{c}{a}$  = 0. The coordinates of the centre are $\left (-\frac{g}{a} , -\frac{f}{a} \right )$ and radius = $\sqrt{\frac{g^{2}}{a^{2}} + \frac{f^{2}}{a^{2}} - \frac{c}{a}}$

5. On comparing the equation x2 + y2 + 2gx + 2fy + c = 0 of a circle with general equation of a second degree ax2 + by2 + 2hxy + 2gx + 2fy + c = 0, we find that it represents a circle if a = b, and h = 0

PROPERTIES OF CIRCLE

Some particular cases of circle: The equation of a circle at (h, k) and radius equal to a, is (x - h)2 + (y - k)2 = a2 ---- (i)

(i) When the centre of the circle coincides with the origin

Circle Coincides with the Origin

                                h = k = 0
Then, equation (i) reduces to x2 + y2 = a2

(ii) When the circle passes through the origin

Let O be the origin and C (h, k) be the centre of the circle. Draw CM  OX.

Circle Passes Through the Origin

In triangle OCM, OC2 = OM2 + CM2
                            a2 = h2 + k2
The equation of the circle (i) then becomes
(x - h)2 + (y - k)2 = h2 + k2
x2 + y2 -2hx -2ky = 0

(iii) When the circle touches x- axis

Let C (h, k) be the centre of the circle, since the circle touches the x-axis

Circle Touches X Axis

Therefore, a = k
Hence, the equation of the circle is
(x - h)2 + (y - a)2 = a2
    x2 + y2 -2hx -2ay + h2 = 0

(iv)  When the circle touches y- axis

Let C (h, k) be the centre of the circle, since the circle touches the y-axis

Circle Touches Y Axis

Therefore, a = h
Hence, the equation of the circle is
(x - a)2 + (y - k)2 = a2
    x2 + y2 -2ax -2ky + k2 = 0

(v) When the circle touches both the axes

Let C (h, k) be the centre of the circle, since the circle touches the x-axis

Circle Touches Both Axis

Therefore, a = h = k.
Hence, the equation of the circle is
(x - a)2 + (y - a)2 = a2
x2 + y2 -2ax -2ay + a2 = 0

(vi) When the circle passes through the origin and the centre lies on the x-axis

In this case, we have k =0 and h = a

Circle Centre Lies On X Axis

Hence, the equation of the circle is
(x - a)2 + (y - 0)2 = a2
       x2 + y2 - 2ax = 0

(vii) When the circle passes through the origin and the centre lies on the y-axis

In this case, we have h =0 and k = a

Circlec Centre Lies On Y Axis

Hence, the equation of the circle is
(x - 0)2 + (y - a)2 = a2
        x2 + y2 - 2ay = 0

Parabola

The Parabola Definition:A parabola is the locus of a point which moves in a plane such that its distance from a fixed a point in the plane is always equal to its distance from a fixed straight.

Conic Sections in General Form:  ax2 + by2 + cx + dy + e = 0

If either a = 0 or b = 0 then the equation defines a parabola (x2 or y2 is missing).  Isolate the variable that is not squared and use the completing the square method to convert the equation to that of a Parabola in Standard Form.
The standard form of the equation of a parabola with vertex (h, k) and axis of symmetry x = h is y = a(x - h)2 +k.

  • If a is positive, then graph opens upward and has a minimum.
  • If a is negative, then graph opens downward and has a maximum.       
A few standard forms of Parabola:

1. The Standard form of a horizontal parabola that opens to the right and has a vertex at (0, 0) is y2 = 4px

Standard Form Of Parabola

Properties of the above Horizontal Parabola:

(i) p is the distance from the vertex of the parabola to the focus or directrix
(ii) This makes the coordinates of the focus (p, 0)
(iii) This makes the equation of the directrix x = - p
(iv) The makes the axis of symmetry the x-axis (y = 0)

2.  The Standard Form of a Parabola that opens to the left and has a vertex at (0, 0) is  y2 = - 4px

Parabola Example Problem

Properties of the above Horizontal parabola:

(i) p is the distance from the vertex of the parabola to the focus or directrix
(ii) This makes the coordinates of the focus(- p, 0)
(iii) This makes the equation of the directrix x = p
(iv) The makes the axis of symmetry the x-axis (y = 0)

3. The Standard Form of a Parabola that opens up and has a vertex at (0, 0) is x2 = 4ay

Standard Form of Parabola

Properties of the Parabola:

(i) p is the distance from the vertex of the parabola to the focus or directrix
(ii) This makes the coordinates of the focus (0,p)
(iii) This makes the equation of the directrix  y = -p
(iv) This makes the axis of symmetry the y-axis (x = 0)

4.The Standard Form of a Parabola that opens down and has a vertex at   (0, 0) is x2 = - 4py

Parabola in x Axis

Properties of the above Parabola:

(i) p is the distance from the vertex of the parabola to the focus or directrix
(ii) This makes the coordinates of the focus (0,- p)
(iii) This makes the equation of the directrix  y = p
(iv) This makes the axis of symmetry the y-axis (x = 0)

Parabola Formula:

 S.No   y2 = 4ax  y2 =-  4ax  x2 = 4ay  x2 = - 4ay
 1  Coordinates of vertex   (0,0)  (0,0)  (0,0)  (0,0)
 2  Coordinates of focus  (a,0)  (-a,0)  (0,a)  (0,-a)
 3  Equation of the directrix   x=-a  x=a  y=-a  y=a
 4  Equation of the axis  y=0  y=0  x=0  x=0
 5  Length of the Latus - rectum   4a  4a  4a  4a
 6  Focal distance of the point (x, y)   a+x  a-x  a+y  a-y

Conic Problems


Ellipse

As always, the eccentricity is defined as the ratio of the distance of any point on the ellipse from the focus to its distance from the corresponding directrix. It is denoted by ‘e’.

Ellipse

The relation between a, b and e is given as

e = $\frac{\sqrt{a^{2}-b^{2}}}{a}$

This equation is applicable for an ellipse whose semi major axis is a and semi minor axis is b.
Alternately, eccentricity can be defined as the ratio of the distance of the focus from the centre c to the distance of the vertex from the centre a.
                e = $\frac{c}{a}$

e for an ellipse is always less than 1
   $\frac{x^{2}}{a^{2}}$ + $\frac{y^{2}}{b^{2}}$ = 1 , a>b    $\frac{x^{2}}{a^{2}}$ + $\frac{y^{2}}{b^{2}}$ = 1 , a<b 
 Coordinates of the centre  (0,0)  (0,0)
 Coordinates of the vertices  (a,0) and (-a,0)  (0,b) and (0,-b)
 Coordinates of foci  (ae,0) and (-ae,0)  (0,be) and (0,-be)
 Length of Major axis  2a  2b
 Length of Minor axis  2b  2a
 Equation of Major axis  y=0 x=0
 Equation of Minor axis  x=0  y=0
 Equation of Directrices  x = $\frac{a}{e}$  and x = -$\frac{a}{e}$  x = $\frac{b}{e}$ and x = -$\frac{b}{e}$
 Eccentricity                 
 e = $\sqrt{1-\frac{b^{2}}{a^{2}}}$  e = $\sqrt{1-\frac{a^{2}}{b^{2}}}$
 Length of Latus rectum  $\frac{2b^{2}}{a^{2}}$  $\frac{2a^{2}}{b^{2}}$
 Focal distance of point (x , y)  a $\pm$ ex  b$\pm$ey


Hyperbola

An Hyperbola is the locus of a point in a plane which moves in the plane in such a way that the ratio of its distance from a fixed point in the same to its distance from a fixed straight line is always constant and greater than unity.

Eccentricity: As always, the eccentricity is defined as the ratio of the distance of any point on the ellipse from the focus to its distance from the corresponding directrix. It is denoted by 'e'.

Vertices: The points A and A’ where the curve meets the line joining foci S and S’ are called the vertices of the hyperbola.

Foci: The points S (ae, 0) and S’ (- ae, 0) are the foci of the hyperbola.

Transverse and Conjugate axis: The straight line joining the vertices A and A’ is called the transverse axis of the hyperbola. Its length AA’ is generally taken to be “2a”.

Latus-Rectum:
LSL' is the latusrectum and LS is called the semi latusrectum. TST’ is also a latusrectum.

Directrices:
ZK and Z’K’ are called the directrices of the hyperbola and their equations are x = $\frac{a}{e}$ and x = - $\frac{a}{e}$

Centre: The point C (0, 0) is called the centre of the circle.

Focal Distances:
The distance of any point on the hyperbola from its foci is known as its focal distances.

Conjugate Hyperbola: The Hyperbola whose transverse and conjugate axes are respectively the conjugate and transverse axes of a given hyperbola is called the conjugate hyperbola of the given hyperbola.

‘e’ for an Hyperbola is always greater than 1
Special Form: If the centre of the hyperbola is at a point (h, k) and the directions of the axes are parallel to the coordinate axes then its equation is

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

  $\frac{x^{2}}{a^{2}}$ - $\frac{y^{2}}{b^{2}}$ = 1
Hyperbola
 -$\frac{x^{2}}{a^{2}}$ + $\frac{y^{2}}{b^{2}}$ = 1
Conjugate Hyperbola
Coordinates of the centre   ( 0 , 0 )  ( 0 , 0 )
 Coordinates of the vertices  (a , 0 ) and ( -a , 0 )  ( 0 , b ) and ( 0 , - b )
Coordinates of foci  ( ae , 0 ) and ( -ae , 0 )  ( 0 , be ) and ( 0 , -be )
 Length of Transverse axis  2a  2b
Length of Conjugate axis  2b  2a
 Equation of Transverse axis  y = 0  x = 0
 Equation of Conjugate axis  x = 0  y = 0
 Equation of Directrices
 x = $\frac{a}{e}$ and x = -$\frac{a}{e}$
 x = $\frac{b}{e}$ and x = -$\frac{b}{e}$
 Eccentricity  e = $\sqrt{\frac{a^{2}+b^{2}}{a^{2}}}$  e = -$\sqrt{\frac{b^{2}+a^{2}}{b^{2}}}$
 Length of Latusrectum
 $\frac{2b^{2}}{a^{2}}$
 $\frac{2a^{2}}{b^{2}}$
 Focal distance of point (x , y)  ex$\pm$a  ey$\pm$b

Topics in Conics