Detailed Notes: Parabola | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Page 1


PARABOLA 
1. CONIC SECTIONS 
A conic section, or conic is the locus of a point which moves in a plane so that its distance from a 
fixed point is in a constant ratio to its perpendicular distance from a fixed straight line. 
(a) The fixed point is called the focus. 
(b) The fixed straight line is called the directrix. 
(c) The constant ratio is called the eccentricity denoted by e. 
(d) The line passing through the focus & perpendicular to the directrix is called the axis. 
(e) A point of intersection of a conic with its axis is called a vertex. 
2. GENERAL EQUATION OF A CONIC: FOCAL DIRECTRIX PROPERTY: 
The general equation of a conic with focus ( ?? , ?? ) & directrix ???? + ???? + ?? = 0 is: 
( ?? 2
+ ?? 2
) [( ?? - ?? )
2
+ ( ?? - ?? )
2
] = ?? 2
( ???? + ???? + ?? )
2
= ?? ?? 2
+ 2h???? + ?? ?? 2
+ 2???? + 2???? + ?? = 0 
3. DISTINGUISHING BETWEEN THE CONIC 
The nature of the conic section depends upon the position of the focus ?? w.r.t. the directrix & also 
upon the value of the eccentricity e. Two different cases arise. 
Case (i) When the focus lies on the directrix: 
In this case ?? = ?????? + 2???? h - ????
2
- ????
2
- ?? h
2
= 0& the general equation of a conic represents a 
pair of straight lines and if: 
?? > 1 the lines will be real & distinct intersecting at ?? . 
?? = 1 the lines will coincident. 
?? < 1 the lines will be imaginary. 
Case (ii) When the focus does not lie on the directrix: 
The conic represents: 
a parabola an ellipse a hyperbola a rectangular hyperbola 
?? = 1; ?? ? 0 0 < ?? < 1; ?? ? 0 ?? ? 0; ?? > 1; ?? > 1; ?? ? 0 
 h
2
= ???? h
2
< ???? 
h
2
> ???? ; ?? + ?? = 0 
 
 
Page 2


PARABOLA 
1. CONIC SECTIONS 
A conic section, or conic is the locus of a point which moves in a plane so that its distance from a 
fixed point is in a constant ratio to its perpendicular distance from a fixed straight line. 
(a) The fixed point is called the focus. 
(b) The fixed straight line is called the directrix. 
(c) The constant ratio is called the eccentricity denoted by e. 
(d) The line passing through the focus & perpendicular to the directrix is called the axis. 
(e) A point of intersection of a conic with its axis is called a vertex. 
2. GENERAL EQUATION OF A CONIC: FOCAL DIRECTRIX PROPERTY: 
The general equation of a conic with focus ( ?? , ?? ) & directrix ???? + ???? + ?? = 0 is: 
( ?? 2
+ ?? 2
) [( ?? - ?? )
2
+ ( ?? - ?? )
2
] = ?? 2
( ???? + ???? + ?? )
2
= ?? ?? 2
+ 2h???? + ?? ?? 2
+ 2???? + 2???? + ?? = 0 
3. DISTINGUISHING BETWEEN THE CONIC 
The nature of the conic section depends upon the position of the focus ?? w.r.t. the directrix & also 
upon the value of the eccentricity e. Two different cases arise. 
Case (i) When the focus lies on the directrix: 
In this case ?? = ?????? + 2???? h - ????
2
- ????
2
- ?? h
2
= 0& the general equation of a conic represents a 
pair of straight lines and if: 
?? > 1 the lines will be real & distinct intersecting at ?? . 
?? = 1 the lines will coincident. 
?? < 1 the lines will be imaginary. 
Case (ii) When the focus does not lie on the directrix: 
The conic represents: 
a parabola an ellipse a hyperbola a rectangular hyperbola 
?? = 1; ?? ? 0 0 < ?? < 1; ?? ? 0 ?? ? 0; ?? > 1; ?? > 1; ?? ? 0 
 h
2
= ???? h
2
< ???? 
h
2
> ???? ; ?? + ?? = 0 
 
 
4. PARABOLA: 
A parabola is the locus of a point which moves in a plane, such that its distance from a fixed point 
(focus) is equal to its perpendicular distance from a fixed straight line (directrix). 
Standard equation of a parabola is ?? 2
= 4???? . For this parabola: 
(i) Vertex is ( 0,0 ) 
(ii) Focus is ( ?? , 0 ) 
(iii) Axis is ?? = 0 
(iv) Directrix is ?? + ?? = 0 
(a) Focal distance: 
The distance of a point on the parabola from the focus is called the focal distance of the point. 
(b) Focal chord: 
A chord of the parabola, which passes through the focus is called a focal chord. 
(c) Double ordinate: 
A chord of the parabola perpendicular to the axis of the symmetry is called a double ordinate. 
(d) Latus rectum: 
A double ordinate passing through the focus or a focal chord perpendicular to the axis of parabola is 
called the latus rectum. For ?? 2
= 4???? . 
? Length of the latus rectum = 4?? . 
? Length of the semi latus rectum = 2?? . 
? Ends of the latus rectum are ?? ( ?? , 2?? ) &?? '( ?? , -2?? ) 
Note that: 
(i) Perpendicular distance from focus on directrix = half the latus rectum. 
(ii) Vertex is middle point of the focus & the point of intersection of directrix & axis. 
(iii) Two parabolas are said to be equal if they have the same latus rectum. 
5. PARAMETRIC REPRESENTATION: 
The simplest & the best form of representing the co-ordinates of a point on the parabola is (at  
2
, 
2at) 
. The equation ?? = ????
2
&?? = 2at together represents the parabola ?? 2
= 4???? , ?? being the parameter. 
 
 
Page 3


PARABOLA 
1. CONIC SECTIONS 
A conic section, or conic is the locus of a point which moves in a plane so that its distance from a 
fixed point is in a constant ratio to its perpendicular distance from a fixed straight line. 
(a) The fixed point is called the focus. 
(b) The fixed straight line is called the directrix. 
(c) The constant ratio is called the eccentricity denoted by e. 
(d) The line passing through the focus & perpendicular to the directrix is called the axis. 
(e) A point of intersection of a conic with its axis is called a vertex. 
2. GENERAL EQUATION OF A CONIC: FOCAL DIRECTRIX PROPERTY: 
The general equation of a conic with focus ( ?? , ?? ) & directrix ???? + ???? + ?? = 0 is: 
( ?? 2
+ ?? 2
) [( ?? - ?? )
2
+ ( ?? - ?? )
2
] = ?? 2
( ???? + ???? + ?? )
2
= ?? ?? 2
+ 2h???? + ?? ?? 2
+ 2???? + 2???? + ?? = 0 
3. DISTINGUISHING BETWEEN THE CONIC 
The nature of the conic section depends upon the position of the focus ?? w.r.t. the directrix & also 
upon the value of the eccentricity e. Two different cases arise. 
Case (i) When the focus lies on the directrix: 
In this case ?? = ?????? + 2???? h - ????
2
- ????
2
- ?? h
2
= 0& the general equation of a conic represents a 
pair of straight lines and if: 
?? > 1 the lines will be real & distinct intersecting at ?? . 
?? = 1 the lines will coincident. 
?? < 1 the lines will be imaginary. 
Case (ii) When the focus does not lie on the directrix: 
The conic represents: 
a parabola an ellipse a hyperbola a rectangular hyperbola 
?? = 1; ?? ? 0 0 < ?? < 1; ?? ? 0 ?? ? 0; ?? > 1; ?? > 1; ?? ? 0 
 h
2
= ???? h
2
< ???? 
h
2
> ???? ; ?? + ?? = 0 
 
 
4. PARABOLA: 
A parabola is the locus of a point which moves in a plane, such that its distance from a fixed point 
(focus) is equal to its perpendicular distance from a fixed straight line (directrix). 
Standard equation of a parabola is ?? 2
= 4???? . For this parabola: 
(i) Vertex is ( 0,0 ) 
(ii) Focus is ( ?? , 0 ) 
(iii) Axis is ?? = 0 
(iv) Directrix is ?? + ?? = 0 
(a) Focal distance: 
The distance of a point on the parabola from the focus is called the focal distance of the point. 
(b) Focal chord: 
A chord of the parabola, which passes through the focus is called a focal chord. 
(c) Double ordinate: 
A chord of the parabola perpendicular to the axis of the symmetry is called a double ordinate. 
(d) Latus rectum: 
A double ordinate passing through the focus or a focal chord perpendicular to the axis of parabola is 
called the latus rectum. For ?? 2
= 4???? . 
? Length of the latus rectum = 4?? . 
? Length of the semi latus rectum = 2?? . 
? Ends of the latus rectum are ?? ( ?? , 2?? ) &?? '( ?? , -2?? ) 
Note that: 
(i) Perpendicular distance from focus on directrix = half the latus rectum. 
(ii) Vertex is middle point of the focus & the point of intersection of directrix & axis. 
(iii) Two parabolas are said to be equal if they have the same latus rectum. 
5. PARAMETRIC REPRESENTATION: 
The simplest & the best form of representing the co-ordinates of a point on the parabola is (at  
2
, 
2at) 
. The equation ?? = ????
2
&?? = 2at together represents the parabola ?? 2
= 4???? , ?? being the parameter. 
 
 
6. TYPE OF PARABOLA: 
Four standard forms of the parabola are ?? 2
= 4???? ; ?? 2
= -4???? ; ?? 2
= 4???? ; ?? 2
= -4???? 
 
?? 2
= 4???? 
 
?? 2
= -4???? 
Page 4


PARABOLA 
1. CONIC SECTIONS 
A conic section, or conic is the locus of a point which moves in a plane so that its distance from a 
fixed point is in a constant ratio to its perpendicular distance from a fixed straight line. 
(a) The fixed point is called the focus. 
(b) The fixed straight line is called the directrix. 
(c) The constant ratio is called the eccentricity denoted by e. 
(d) The line passing through the focus & perpendicular to the directrix is called the axis. 
(e) A point of intersection of a conic with its axis is called a vertex. 
2. GENERAL EQUATION OF A CONIC: FOCAL DIRECTRIX PROPERTY: 
The general equation of a conic with focus ( ?? , ?? ) & directrix ???? + ???? + ?? = 0 is: 
( ?? 2
+ ?? 2
) [( ?? - ?? )
2
+ ( ?? - ?? )
2
] = ?? 2
( ???? + ???? + ?? )
2
= ?? ?? 2
+ 2h???? + ?? ?? 2
+ 2???? + 2???? + ?? = 0 
3. DISTINGUISHING BETWEEN THE CONIC 
The nature of the conic section depends upon the position of the focus ?? w.r.t. the directrix & also 
upon the value of the eccentricity e. Two different cases arise. 
Case (i) When the focus lies on the directrix: 
In this case ?? = ?????? + 2???? h - ????
2
- ????
2
- ?? h
2
= 0& the general equation of a conic represents a 
pair of straight lines and if: 
?? > 1 the lines will be real & distinct intersecting at ?? . 
?? = 1 the lines will coincident. 
?? < 1 the lines will be imaginary. 
Case (ii) When the focus does not lie on the directrix: 
The conic represents: 
a parabola an ellipse a hyperbola a rectangular hyperbola 
?? = 1; ?? ? 0 0 < ?? < 1; ?? ? 0 ?? ? 0; ?? > 1; ?? > 1; ?? ? 0 
 h
2
= ???? h
2
< ???? 
h
2
> ???? ; ?? + ?? = 0 
 
 
4. PARABOLA: 
A parabola is the locus of a point which moves in a plane, such that its distance from a fixed point 
(focus) is equal to its perpendicular distance from a fixed straight line (directrix). 
Standard equation of a parabola is ?? 2
= 4???? . For this parabola: 
(i) Vertex is ( 0,0 ) 
(ii) Focus is ( ?? , 0 ) 
(iii) Axis is ?? = 0 
(iv) Directrix is ?? + ?? = 0 
(a) Focal distance: 
The distance of a point on the parabola from the focus is called the focal distance of the point. 
(b) Focal chord: 
A chord of the parabola, which passes through the focus is called a focal chord. 
(c) Double ordinate: 
A chord of the parabola perpendicular to the axis of the symmetry is called a double ordinate. 
(d) Latus rectum: 
A double ordinate passing through the focus or a focal chord perpendicular to the axis of parabola is 
called the latus rectum. For ?? 2
= 4???? . 
? Length of the latus rectum = 4?? . 
? Length of the semi latus rectum = 2?? . 
? Ends of the latus rectum are ?? ( ?? , 2?? ) &?? '( ?? , -2?? ) 
Note that: 
(i) Perpendicular distance from focus on directrix = half the latus rectum. 
(ii) Vertex is middle point of the focus & the point of intersection of directrix & axis. 
(iii) Two parabolas are said to be equal if they have the same latus rectum. 
5. PARAMETRIC REPRESENTATION: 
The simplest & the best form of representing the co-ordinates of a point on the parabola is (at  
2
, 
2at) 
. The equation ?? = ????
2
&?? = 2at together represents the parabola ?? 2
= 4???? , ?? being the parameter. 
 
 
6. TYPE OF PARABOLA: 
Four standard forms of the parabola are ?? 2
= 4???? ; ?? 2
= -4???? ; ?? 2
= 4???? ; ?? 2
= -4???? 
 
?? 2
= 4???? 
 
?? 2
= -4???? 
 
?? 2
= 4???? 
 
 
 
 
 
?? 2
= -4???? 
 
Page 5


PARABOLA 
1. CONIC SECTIONS 
A conic section, or conic is the locus of a point which moves in a plane so that its distance from a 
fixed point is in a constant ratio to its perpendicular distance from a fixed straight line. 
(a) The fixed point is called the focus. 
(b) The fixed straight line is called the directrix. 
(c) The constant ratio is called the eccentricity denoted by e. 
(d) The line passing through the focus & perpendicular to the directrix is called the axis. 
(e) A point of intersection of a conic with its axis is called a vertex. 
2. GENERAL EQUATION OF A CONIC: FOCAL DIRECTRIX PROPERTY: 
The general equation of a conic with focus ( ?? , ?? ) & directrix ???? + ???? + ?? = 0 is: 
( ?? 2
+ ?? 2
) [( ?? - ?? )
2
+ ( ?? - ?? )
2
] = ?? 2
( ???? + ???? + ?? )
2
= ?? ?? 2
+ 2h???? + ?? ?? 2
+ 2???? + 2???? + ?? = 0 
3. DISTINGUISHING BETWEEN THE CONIC 
The nature of the conic section depends upon the position of the focus ?? w.r.t. the directrix & also 
upon the value of the eccentricity e. Two different cases arise. 
Case (i) When the focus lies on the directrix: 
In this case ?? = ?????? + 2???? h - ????
2
- ????
2
- ?? h
2
= 0& the general equation of a conic represents a 
pair of straight lines and if: 
?? > 1 the lines will be real & distinct intersecting at ?? . 
?? = 1 the lines will coincident. 
?? < 1 the lines will be imaginary. 
Case (ii) When the focus does not lie on the directrix: 
The conic represents: 
a parabola an ellipse a hyperbola a rectangular hyperbola 
?? = 1; ?? ? 0 0 < ?? < 1; ?? ? 0 ?? ? 0; ?? > 1; ?? > 1; ?? ? 0 
 h
2
= ???? h
2
< ???? 
h
2
> ???? ; ?? + ?? = 0 
 
 
4. PARABOLA: 
A parabola is the locus of a point which moves in a plane, such that its distance from a fixed point 
(focus) is equal to its perpendicular distance from a fixed straight line (directrix). 
Standard equation of a parabola is ?? 2
= 4???? . For this parabola: 
(i) Vertex is ( 0,0 ) 
(ii) Focus is ( ?? , 0 ) 
(iii) Axis is ?? = 0 
(iv) Directrix is ?? + ?? = 0 
(a) Focal distance: 
The distance of a point on the parabola from the focus is called the focal distance of the point. 
(b) Focal chord: 
A chord of the parabola, which passes through the focus is called a focal chord. 
(c) Double ordinate: 
A chord of the parabola perpendicular to the axis of the symmetry is called a double ordinate. 
(d) Latus rectum: 
A double ordinate passing through the focus or a focal chord perpendicular to the axis of parabola is 
called the latus rectum. For ?? 2
= 4???? . 
? Length of the latus rectum = 4?? . 
? Length of the semi latus rectum = 2?? . 
? Ends of the latus rectum are ?? ( ?? , 2?? ) &?? '( ?? , -2?? ) 
Note that: 
(i) Perpendicular distance from focus on directrix = half the latus rectum. 
(ii) Vertex is middle point of the focus & the point of intersection of directrix & axis. 
(iii) Two parabolas are said to be equal if they have the same latus rectum. 
5. PARAMETRIC REPRESENTATION: 
The simplest & the best form of representing the co-ordinates of a point on the parabola is (at  
2
, 
2at) 
. The equation ?? = ????
2
&?? = 2at together represents the parabola ?? 2
= 4???? , ?? being the parameter. 
 
 
6. TYPE OF PARABOLA: 
Four standard forms of the parabola are ?? 2
= 4???? ; ?? 2
= -4???? ; ?? 2
= 4???? ; ?? 2
= -4???? 
 
?? 2
= 4???? 
 
?? 2
= -4???? 
 
?? 2
= 4???? 
 
 
 
 
 
?? 2
= -4???? 
 
Parabola Vertex Focus Axis Directrix Length 
of 
Latus 
rectum 
Ends of 
Latus 
rectum 
Parametric 
equation 
Focal 
length 
?? 2
= 4???? ( 0,0) ( ?? , 0) ?? = 0 ?? = -?? 4?? ( ?? , ±2?? ) ( ????
2
, 2???? ) ?? + ?? 
?? 2
= -4???? ( 0,0) ( -?? , 0) ?? = 0 ?? = ?? 4?? ( -?? , ±2?? ) ( -????
2
, 2???? ) ?? - ?? 
?? 2
= +4???? ( 0,0) ( 0, ?? ) ?? = 0 ?? = -?? 4a ( ±2?? , ?? ) ( 2???? , ?? ?? 2
) ?? + ?? 
?? 2
= -4???? ( 0,0) ( 0, -?? ) ?? = 0 ?? = ?? 4a ( ±2?? , -?? ) ( 2???? , -????
2
) ?? - ?? 
( ?? - ?? )
2
= 4?? ( ?? - h) 
( h, ?? ) ( h
+ ?? , ?? ) 
?? = ?? ?? + ?? - h
= 0 
4a ( h + ?? , ?? ± 2?? ) 
( h + ?? ?? 2
, ?? + 2???? ) 
?? - h
+ ?? 
( ?? - ?? )
2
= 4?? ( ?? - ?? ) 
( ?? , ?? ) ( ?? , ?? + ?? ) 
?? = ?? ?? + ?? - ?? = 0 
4?? ( ?? ± 2?? , ?? + ?? ) 
( ?? + 2???? , ?? + ?? ?? 2
) 
?? - ?? + ?? 
 
Problem 1: Find the vertex, axis, directrix, focus, latus rectum and the tangent at vertex for the 
parabola 9?? 2
- 16?? - 12?? - 57 = 0. 
Solution: The given equation can be rewritten as (?? -
2
3
)
2
=
16
9
(?? +
61
16
) which is of the form ?? 2
=
4???? . Hence the vertex is (-
61
16
,
2
3
) 
The axis is ?? -
2
3
= 0 ? ?? =
2
3
 
The directrix is ?? + ?? = 0 ? ?? +
61
16
+
4
9
= 0 ? ?? = -
613
144
 
The focus is ?? = ?? and ?? = 0 ? ?? +
61
16
=
4
9
 and ?? -
2
3
= 0 
? focus = (-
485
144
,
2
3
) 
Length of the latus rectum = 4 ?? =
16
9
 
The tangent at the vertex is ?? = 0 ? ?? = -
61
16
. 
Ans. 
Problem 2: The length of latus rectum of a parabola, whose focus is ( 2,3) and directrix is the line ?? -
4?? + 3 = 0 is - 
(A) 
7
v 17
 
(B) 
14
v 21