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.
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.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 semilatus 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:
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
Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0, with A, B, C not all simultaneously 0.We can then find the type of conic from the sign of B
^{2}  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
10x
^{2} + 9y
^{2} + 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
B
^{2} – 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
y
^{2} + (1 – e
^{2})x
^{2} – 2px + p
^{2 }= 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

x^{2} + y^{2} = r^{2}
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

y^{2} = 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) 
Semilatus rectum (e) 
Focal parameter (p) 
Circle 
x^{2} + y^{2} = r^{2 } 
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 
y^{2}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 SECTIONSDefinition: 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} = r
^{2}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 xaxis.
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 = at
^{2}A variable point on the parabola is given by (2at, at
^{2}), 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
$\frac{(xh)^{2}}{a^{2}}$ +
$\frac{(yk)^{2}}{b^{2}}$ = 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
$\frac{(xh)^{2}}{a^{2}}$ 
$\frac{(yk)^{2}}{b^{2}}$ = 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.
In a polar equation of a conic, the pole is the focus of the conic, and the polar axis lies along the positive xaxis. Let â€˜pâ€™ 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.
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.(kr$\cos \theta$)
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
 r = $\frac{ek}{1 + e.\cos \theta}$ , if the directrix is the line x = k > 0.
 r = $\frac{ek}{1 + e.\cos \theta}$ , if the directrix is the line x = k < 0.
 r = $\frac{ek}{1 + e.\sin \theta}$ , if the directrix is the line y = k > 0.
 r = $\frac{ek}{1 + e.\sin \theta}$ ,if the directrix is the line x = k < 0.
Graphing Conics
Graphing a parabolaGraphing the parabola y
^{2} = 4ax, a > 0
The given equation y
^{2}= 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: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 = r
^{2} (x  h)
^{2} + (y  k)
^{2} = r
^{2}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 x^{2} + y^{2} = r^{2}
General Equation of a circle:The equation x
^{2} + y
^{2} + 2gx + 2fy + c = 0 always represents a circle whose centre is
( g,  f) and radius
Proof:
The given equation is x
^{2} + y
^{2} + 2gx + 2fy + c = 0
=> (x
^{2} + 2gx + g
^{2} )+ (y
^{2} + 2fy + f
^{2}) = g
^{2} + f
^{2} + 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} = r
^{2} which represents a circle having centre (h, k) and radius r.
Therefore clearly, the equation x
^{2} + y
^{2 }= 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 ax^{2} + ay^{2} + 2gx + 2fy + c = 0, a $\neq$ 0, also represents circle. This equation can also be written as ax^{2} + ay^{2} + $\frac{2g}{a}$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 x^{2} + y^{2} + 2gx + 2fy + c = 0 of a circle with general equation of a second degree ax^{2} + by^{2} + 2hxy + 2gx + 2fy + c = 0, we find that it represents a circle if a = b, and h = 0
PROPERTIES OF CIRCLESome particular cases of circle: The equation of a circle at (h, k) and radius equal to a, is (x  h)
^{2} + (y  k)
^{2} = a
^{2}  (i)
(i) When the centre of the circle coincides with the origin h = k = 0
Then, equation (i) reduces to x
^{2} + y
^{2} = a
^{2}(ii) When the circle passes through the originLet O be the origin and C (h, k) be the centre of the circle. Draw CM OX.
In triangle OCM, OC
^{2} = OM
^{2} + CM
^{2} a
^{2} = h
^{2} + k
^{2}The equation of the circle (i) then becomes
(x  h)
^{2} + (y  k)
^{2} = h
^{2} + k
^{2}x
^{2 }+ y
^{2} 2hx 2ky = 0
(iii) When the circle touches x axisLet C (h, k) be the centre of the circle, since the circle touches the xaxis
Therefore, a = k
Hence, the equation of the circle is
(x  h)
^{2} + (y  a)
^{2} = a
^{2} x
^{2} + y
^{2} 2hx 2ay + h
^{2} = 0
(iv) When the circle touches y axisLet C (h, k) be the centre of the circle, since the circle touches the yaxis
Therefore, a = h
Hence, the equation of the circle is
(x  a)
^{2} + (y  k)
^{2} = a
^{2} x
^{2} + y
^{2} 2ax 2ky + k
^{2} = 0
(v) When the circle touches both the axesLet C (h, k) be the centre of the circle, since the circle touches the xaxis
Therefore, a = h = k.
Hence, the equation of the circle is
(x  a)
^{2} + (y  a)
^{2} = a
^{2}x
^{2} + y
^{2} 2ax 2ay + a
^{2} = 0
(vi) When the circle passes through the origin and the centre lies on the xaxisIn this case, we have k =0 and h = a
Hence, the equation of the circle is
(x  a)
^{2} + (y  0)
^{2} = a
^{2} x
^{2} + y
^{2}  2ax = 0
(vii) When the circle passes through the origin and the centre lies on the yaxisIn this case, we have h =0 and k = a
Hence, the equation of the circle is
(x  0)
^{2} + (y  a)
^{2} = a
^{2} x
^{2} + y
^{2}  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: ax
^{2} + by
^{2} + cx + dy + e = 0
If either a = 0 or b = 0 then the equation defines a parabola (x
^{2} or y
^{2} 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
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 xaxis (y = 0)
2. The Standard Form of a Parabola that opens to the left and has a vertex at (0, 0) is y
^{2} =  4px
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 xaxis (y = 0)
3. The Standard Form of a Parabola that opens up and has a vertex at (0, 0) is x
^{2} = 4ay
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 yaxis (x = 0)
4.The Standard Form of a Parabola that opens down and has a vertex at (0, 0) is x
^{2} =  4py
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 yaxis (x = 0)
Parabola Formula:
S.No


y^{2} = 4ax 
y^{2} = 4ax 
x^{2} = 4ay 
x^{2} =  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 
ax 
a+y 
ay 
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 â€˜eâ€™.
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”.
LatusRectum: 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 1Special 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
$\frac{(xh)^{2}}{a^{2}}$ 
$\frac{(yk)^{2}}{b^{2}}$ = 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 