mathematics. parabola - session 1 session objectives definition of conic section eccentricity...
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Mathematics
PARABOLA - SESSION 1
Session Objectives
• Definition of Conic Section
• Eccentricity
• Definition of Special Points
• Condition for Second degree equation to represent different conic sections
• Standard Form of parabola
• General Form of parabola
• Algorithm for finding special points/ lines
Definition of Conic section
Geometrical Definition
Cross section formed when right circular cone is intersected by a plane
Axis Generator
Circle
Geometrical Definition
Cross section formed when right circular cone is intersected by a plane
Circle
If plane is perpendicular to the axis
Ellipse
Geometrical Definition
Cross section formed when right circular cone is intersected by a plane
Ellipse
• If plane is not perpendicular to the axis
• Does not pass through base
Parabola
Geometrical Definition
Cross section formed when right circular cone is intersected by a plane
Parabola
• If plane is parallel to the generator
Hyperbola
Geometrical Definition
Cross section formed when right circular cone is intersected by a plane
Hyperbola
• Two similar cones
• Plane parallel to the axis
Class Exercise
Are the following be a conic section?
If yes, how they can be generated by intersection of cone(s) and plane.
(i) Point
(ii) Pair of straight lines
Class Exercise
Locus Definition
Locus Definition
Locus of a point moves such that
• Ratio of its distance
• Ratio - Eccentricity
• Fixed Point - Focus
• Fixed Line - Line of Directrix
NPPS
EccentricityPN
Fixed Line
S
Fixed Point
• from a fixed point
• & from a fixed line is constant
Eccentricity and Shapes of Conic Section
e = 1 : Parabola
e < 1 : Ellipse
e = 0 : Circle
e > 1 : Hyperbola
Special Points / Lines
Axis :
Line through Focus and perpendicular to line of directrix
Directrix
NP
S
Focus
Axis
Vertex :
Meeting point of Curve and axis
Vertex
Special Points / Lines
Double Ordinate :
Line segment joint two points on a conic for one particular value of abscissa
Latus rectum :
Double ordinate passing through Focus
Directrix
NP
S
Focus
AxisVertex
Standard Form of Parabola
e =1
• Axis is x- axis , y = 0
• Vertex - ( 0,0)
• Focus - ( a,0)
Directrix
NP
S
Focus
AxisVertex
V
As e = 1 , SV = VV1
V1
Equation of Directrix : x a
2 2PS a 0
Let P be ( , )
2
aPN a
1 0
Standard Form of Parabola
e =1
Directrix
NP
S
Focus
AxisVertex
VV1
PSNow e 1
PN
2 2PS PN a a
2 22a a
2 4a
2Equation of parabola y 4ax
Standard Form of Parabola- Special Point / lines
Directrix
NP
S
Focus
AxisVertex
VV1
2Equation of parabola y 4ax Focus : ( a,0) , Vertex : ( 0,0)
Axis : y = 0 , Directrix : x = – a
P.O.I of this line and Parabola :
y2 = 4a (a)
L
L’
Length of Latus rectum :
Eq. Of SLL’ : x = a
L a,2a L ' a, 2a
y 2a
LL ' 4a
Standard Form of Parabola- Special Point / lines
2Equation of parabola y 4ax Focus : ( -a,0) , Vertex : ( 0,0)
Axis : y = 0 , Directrix : x =–(– a)
P.O.I of this line and Parabola :
y2 = – 4a (–a)
N
S
Focus
Vertex
V
Directrix
P
Axis V1
L
L’
Length of Latus rectum :
Eq. Of SLL’ : x = –a
L a,2a L ' a, 2a
y 2a
LL ' 4a
Standard Form of Parabola- Special Point / lines
2Equation of parabola x 4ay
Focus : ( 0,a) , Vertex : ( 0,0)
Axis : x = 0 , Directrix : y =–( a)
P.O.I of this line and Parabola :
x2 = 4a (a)
Length of Latus rectum :
Eq. Of SLL’ : y = a
L 2a,a L ' 2a,a
x 2a
LL ' 4a
S
Focus
V
Directrix N
P
Axis
V1
LL’
Standard Form of Parabola- Special Point / lines
2Equation of parabola x 4ay
Focus : ( 0,–a) , Vertex : ( 0,0)
Axis : x = 0 , Directrix : y =( a)
P.O.I of this line and Parabola :
x2 = –4a (–a)
Length of Latus rectum :
Eq. Of SLL’ : y = – a
L 2a, a L ' 2a, a
x 2a
LL ' 4a
S
Focus
V
Directrix N
P
Axis
V1
LL’
Algorithm to Find out special points - Standard Form
2 2y 4ax , x 4ay
Vertex : (0,0)
Axis : Put Second degree variable = 0
Focus :
If second degree variable is y : ( a,0)
If second degree variable is x : (0, a)
Line of Directrix :
If second degree variable is y : x = – ( a)
If second degree variable is x : y = – ( a)
Length of Latus rectum : 4a
Class Exercise
Axis : Put Second degree variable = 0
Focus :
If second degree variable is x : (0, a)
Line of Directrix :
If second degree variable is x : y = – ( a)
Length of Latus rectum : 18 units
Find the focus, line of directrix and length of latus rectum for the parabola represented by
2x –18y.
Solution : x = 0
418
0 ,
29
y
29
0 ,
29y
Class Exercise
For what point of parabola y2 = 18 x is the y-coordinate equal to three times the x-coordinate?
Solution :
3,beintpotheLet
As this point is on parabola 183 2
02 , 0062 ,,beintPo
General Form - Parabola
Focus : (x1,y1) ,
Line of directrix : Ax + By + 1 = 0
e =1
2 21 1Dis tance from Focus x y
Let P be ( , )
2 2
A B 1Dis tance from Directrix
A B
2 21 1 2 2
A B 1x y
A B
22 21 1 2 2
A B 1x y
A B
General Form - Parabola
22 21 1 2 2
A B 1x y
A B
2 2 2 2 2 2 2 21 1
2 2 2 21 1
B A 2AB 2 x A B A 2 y A B B
x y A B 1
2 2 2 2B x A y 2ABxy ........
General Form - Parabola
Comparing with
2 2ax by 2hxy 2gx 2fy c 0
2 2a B , b A , h AB
2h ab
2 2 2 2B x A y 2ABxy ........
One of the Condition for second degree equation to represent parabola
Class Test
Class Exercise
?abhthatenoughitis,parabolarepresent
tocfygxhxybyaxFor
2
22 0222
Solution : 02 222 chbgaffghabc
Pre – session - 6
Class Exercise
Solution :
If the focus is (4, 5) and line of
directrix is x + 2y + 1 = 0, the
equation of the parabola will be ?
Let P ,β be the point whose locus be the
desired parabola
2 2Dis tance from Focus 4 5
2 2
2 1Dis tance from Directrix
1 2
Class Exercise
Solution :
If the focus is (4, 5) and line of directrix is
x + 2y + 1 = 0, the equation of the
parabola will be ?
2 2
2 2
4 5e
2 1
1 2
2 2 2
5 4 5 2 1
2 24 4 42 54 204 0
2 24x y 4xy 42x 54y 204 0
General Form - Parabola
2
2
y k x Or
x k y
2I f h ab2 2ax by 2hxy 2gx 2fy c 0
can be converted in to
Algorithm to find Special points/ lines - General Form
2
2
y k x Or
x k y
1. Convert the given equation in to general form
e.g. : y2 – 6y + 24x – 63 = 0
Can be written as : y2 – 6y + 9= – 24x + 72
2y 3 24 x 3
2. Transform the same in to Standard form
2Y 24X , where Y y 3 and X x 3
Algorithm to find Special points/ lines - General Form
3. Find special points/ Line in transformed axis ( X, Y)
4. Reconvert the result in to original axis ( x,y)
2Y 24X , where Y y 3 and X x 3
2For Y 24X ,
Vertex : (0,0), Axis : Y = 0
Focus : (– 6,0) ( as of form y2 = 4ax ) ,
Directrix : X = – (– 6) or X = 6
Vertex : X = 0 x – 3 = 0 x = 3
Y = 0 y – 3 = 0 y = 3 ( 3 ,3)
Focus : ( –3 , 3) , Directrix : x = 9
Class Exercise
2
The focus and the directrix for
the parabola represented by
y 4 8 x 2 is
(a) (0, –4); x = –2 (b) (–4, –2); x = –2(c) (–2, –4); y = –4 (d) (0, –4); x = –4
Transform in to Standard formSolution :
2Y , where Y y 4 and XX x8 3
Find special points/ Line in transformed axis ( X, Y)
Focus - ( 2,0) ; Line of Directrix : X = –2
Class Exercise
2
The focus and the directrix for
the parabola represented by
y 4 8 x 2 is
(a) (0, –4); x = –2 (b) (–4, –2); x = –2(c) (–2, –4); y = –4 (d) (0, –4); x = –4
Solution : Focus - ( 2,0) ; Line of Directrix : X = –2
Reconvert the result in to original axis ( x,y)
For Focus, X 2 2 2 xx 0
Y 0 y y4 0 4 Focus – ( 0 , –4)
Line of directrix is x 2 2 xX 2 4
Practice Exercise - 9
Class Exercise
Solution :
In a parabola , vertex is at (1,1) and line of directrix is x + y = 0. Equation of parabola ?
Axis is y – x = k
Vertex lies on the axis k 1 1 k 0
Axis : y – x = 0
P.O.I of axis and Directrix : (0 , 0)
Let focus be ( h, k) 0 h 0 k
1 h 2 ; 1 k 22 2
Focus – (2 ,2) Vertex is mid point between focus and
P.O.I of axis and directrix
Class Exercise
Solution :
In a parabola , vertex is at (1,1) and line of directrix is x + y = 0. Equation of parabola ?
Focus – (2 ,2) ; Line of directrix : x+y = 0
2 2
Let P , be the point whose locus isdesired parabola
2 21
2
2 2 2 8 8 16 0
2 2Eqn. of parabola : x y 2xy 8x 8y 16 0
Class Exercise
Draw the rough shape of the curve represented by y=ax2+bx+c; where b2– 4ac > 0 , > 0 and b < 0 and find out vertex and axis of parabola.
Compare the results with solution of ax2+bx+c = 0 when b2– 4ac > 0 and a > 0
Transforming the given equation to general form, we get
2y – c bx x
a a
Class Exercise
2y – c bx x
a a
2 22
2 2
b b y c bx 2 x –
2a a a4a 4a
2 2b 1 b – 4acx y
2a a 4a
Transforming the equation into standard form, we get
2 1 bX Y, where X x
a 2a
Shape is parabola
DY y
4a
Class Exercise
2 1 bX Y, where X x
a 2a
–bVertex X coordinate 0 x
2a
Axis: X = 0b
x –2a
b b Dx – Vertex – , –
2a 2a 4a
and a > 0, b < 0, D > 0,
y
x
x = – —b2a
— , b2a
– — D4a
–O
DY y
4a
DY coordinate 0 y
4a
Class Exercise
y=ax2+bx+c
b b Dx – Vertex – , –
2a 2a 4a
and a > 0, b < 0, D > 0,
2 1X Y
a
ax2 + bx + c = 0 (i.e.y = 0) for
two real values of x . ( , )
2ax bx c 0 for x
y
x
x = – —b2a
— , b2a
– — D4a
–O
2ax bx c 0 f or xor x
Thank you