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5/27/2018 Conic Section.pdf
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SYLLABUS JEE MAINS
CONIC SECTION
Sections of cones, equations of conic sections (parabola, ellipse and hyperbola) in standard
forms, condition for y = mx + c to be a tangent and point (s) of tangency.
QUICK REVISION
1. DEFINITION
A parabola is the locus of a point which moves in a plane such that its distance from a
fixed point (called the focus) is equal to its distance from a fixed straight line (called the
directrix).
2. Terms related to parabola
Axis:A straight line passes through the focus and perpendicular to the directrix is called
the axis of parabola.
Vertex :The point of intersection of a parabola and its axis is called the vertex of theparabola.
The vertex is the middle point of the focus and the point of intersection of axis and
directrix.
S(a, 0)
y = 4ax2
L
L'
A
Y
Q
NP(x, y)
X
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Eccentricity :If P be a point on the parabola and PN and PS are the distance from the
directrix and focus S respectively then the ratio PS/PN is called the eccentricity of the
parabola which is denoted by e.
By the definition for the parabola e = 1.
If e > 1 hyperbola, e = 0 circle, e < 1 ellipse
Latus Rectum
Let the given parabola be y2= 4ax. In the figure LSL' (a line through focus to axis) is
the latus rectum.
Also by definition, LSL' = 2 (4a.a)= 4a
Double ordinate
Any chord of the parabola y2= 4ax which is to its axis is called the double ordinate)
through the focus S.
Focal Chord
Any chord to the parabola which passes through the focus is called a focal chord of the
parabola.
3. Some Standard forms of parabola
(1) Parabola opening to left (2) Parabola opening upwards (3) Parabola opening down
wards
(i.e. 2y 4ax ) (a> 0) (i.e. 2x 4ay) ; (a>0) (i.e. 2x 4ay); (a > 0)
yN
Directrix
Vertex
x
+
a=0
Q A
y'
L'
L' (a, -2a)
Latus Rectum
axisx
Focal chord
Double ordinatex = a
L(a, 2a)L
PFocaldistance
Focus
S (a, 0)
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Important terms 42y ax 42y ax 42x ay 42x a
Coordinates of vertex (0, 0) (0, 0) (0, 0) (0, 0)
Coordinates of focus (a, 0) (a, 0) (0, a) (0,a)
Equation of the directrix x a x a y a y= a
Equation of the axis y 0 y 0 x 0 x 0
Length of the latusrectum 4a 4a 4a 4a
Focal distance of a point P(x,y) x a a x y a a y
4. REDUCTION OF STANDARD EQUATION
If the equation of a parabola contains second degree term either in y or in x(but not in
both) then it can be reduced into standard form. For this we change the given equation
into the following forms-
(yk)2= 4a (xh) or (xp)2= 4b (yq)
Then we compare from the following table for the results related to parabola.
Equation of
Parabola
Vertex Axis Focus Directrix Equation
of L.R.
Length
L.R.
2(y K) 4a(x h) (h,k) y k (h a, k ) x a h 0
x a h 4a
2(x p) 4b(y q) (p,q) x p (p,b q ) y b q 0
y b q 4b
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5. PARAMETRIC EQUATIONS OF A PARABOLA
The parametric equation of the parabola y2= 4ax are x = at2, y = 2at, where t is the
parameter.
6. CONDITION FOR TANGENCY AND POINT OF CONTACT
The line y = mx + c touches the parabola y2 = 4ax if c =a
mand the coordinates of the
point of contact are .
Note
The line y = mx + c touches parabola x
2
= 4ay if c =
am
2
The line x cos + y sin = p touches the parabola y2= 4ax if asin2 + p cos =
0.
7. EQUATION OF TANGENT IN DIFFERENT FORMS
7.1 Point Form
The equation of the tangent to the parabola y2= 4ax at the point (x1, y1) is yy1= 2a (x +
x1)
Equation of tangent of all other standard parabolas at (x1, y1)
Equation of parabolas Tangent at (x1, y1)
2y 4ax 1 1yy 2a(x x )
2x 4ay 1 1xx 2a(y y )
2x 4ay 1 1xx 2a(y y )
m
a2,
m
a2
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7.2 Parametric Form
The equation of the tangent to the parabola y2= 4ax at the point (at2, 2at) is ty = x +
at2.
Equations of tangent of all other standard parabolas at ' t'
Equations of
parabolas
Parametric co-ordinates
't'
Tangent at 't'
2y 4ax 2( at ,2at) 2ty x at
2x 4ay 2(2at,at ) 2tx y at
2x 4ay 2(2at, at ) 2tx y at
7.3 Slope Form
The equation of tangent to the parabola y2= 4ax in terms of slope 'm' is y = mx +a
m.
The coordinate of the point of contact are 2a 2a
, mm
Equation of
parabolas
Point of contact in
terms of slope
(m)
Equation of tangent in
terms of slope
(m)
Condition of
Tangency
2y 4ax 2
a 2a,
mm
ay mx
m
ac
m
2y 4ax 2
a 2a,
mm
ay mxm
acm
2x 4ay 2(2am,am ) 2y mx am 2c am
2x 4ay 2( 2am, am ) 2y mx am 2c am
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P(x, y)
Directrix
N
S(focus)
Q
ELLIPSE
1. DEFINITION
An ellipse is the locus of a point which moves in a plane so that the ratio of its distance
from a fixed point (called focus) and a fixed line (called directrix)
is a constant which is less than one. This ratio is called
eccentricity and is denoted by e. For an ellipse, e < 1.
Let S be the focus, QN be the directrix and P be any point on
the ellipse. Then, by definition,PS
PN= e or
PS = e PN, e < 1, where PN is the length of the perpendicular from P on the directrix
QN.
An Alternate DefinitionAn ellipse is the locus of a point that moves in such a way that
the sum of its distacnes from two fixed points (called foci) is constant.
2. EQUATION OF AN ELLIPSE IN STANDARD FORM
The Standard form of the equation of an ellipse is where a and b are
constants.
Symmetry
(a) On replacing y by y, the above equation remains unchanged. So, the curve is
symmetrical about x-axis.
(b) On replacing x by x, the above equation remains unchanged. So, the curve is
symmetrical about y-axis
)ba(1
b
y
a
x2
2
2
2
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3. TERMS RELATED TO AN ELLIPSE
A sketch of the locus of a moving point satisfying the equation , has
been shown in the figure given above.
Foci
If S and S' are the two foci of the ellipse and their coordinates are (ae, 0) and (ae, 0)
respectively, then distance between foci is given by
SS' = 2ae.
Directrices
If ZM and Z' M' are the two directrices of the ellipse and their equations are x =a
eand x
=a
erespectively, then the distance between directrices is given by ZZ' =
2a
e.
Axes The lines AA' and BB' are called the major axis and minor axis respectively of the ellipse.
The length of major axis = AA' = 2a
The length of minor axis = BB' = 2b
B(0,
b)
P(x, y) M
x=a/e
X
ZA(a,0)
Directrix
B(0,-b)
C N
Minor
Axis
Major Axis
N
S'
N'
Z'X'
Directrix
x=-a/e
M'
Y
A'(-a,0)
L
L'
S
P'
)ba(1b
y
a
x2
2
2
2
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CentreThe point of intersection C of the axes of the ellipse is called the centre of the ellipse. All
chords, passing through C are bisected at C.
Vertices The end points A and A' of the major axis are known as the vertices of the ellipse
A (a, 0) and A' (a, 0)
Focal chord A chord of the ellipse passing through its focus is called a focal chord.
Ordinate and Double Ordinate
Let P be a point on the ellipse. From P, draw PN AA' (major axis of the ellipse) and
produce PN to meet the ellipse at P'. Then PN is called an ordinate and PNP' is called
the double ordinate of the point P.
Latus Rectum
If LL' and NN' are the latus rectum of the ellipse, then these lines are to the major axis
AA' passing through the foci S and S' respectively.
Length of latus rectum = LL' =22b
a= NN.
By definition SP = ePM = e = aex and
Thus implies that distances of any point P(x, y) lying on the ellipse from foci are : (a ex)
and (a + ex). In other words
SP + S'P = 2a
i.e., sum of distances of any point P(x, y) lying on the ellipse from foci is constant.
Eccentricity
Since, SP = e.PM, therefore
,a
b,aeL
2
a
b,ae'L
2
a
b,aeN
2
a
b,ae'N
2
xe
a.exax
e
aeP'S
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SP2= e2PM2 or (xae)2+ (y0)2= e2
(xae)2+ y2= (aex)2 x2+ a2e22aex + y2= a22aex +
e2x2
x2(1e2) + y2= a2(1e2)
On comparing with we get
b2= a2(1e2) or e =
Ellipse 2 2
2 2 1
x y
a b
For a> b For b> a
Centre (0, 0) (0, 0)
Vertices ( a,0) (0, b)
Length of major axis 2a 2b
Length of minor axis 2b 2a
Foci ( ae,0) (0, be)
Equation of directrices x a /e y b/ e
Relation in a, band e 2 2 2b a (1 e ) 2 2 2a b (1 e )
Length of latus rectum 22b
a
22a
b
Ends of latus-rectum 2bae,
a
2a
, be
b
Parametric equations (a cos ,bsin ) (a cos ,bsin )(0 2 )
Focal radii 1SP a ex and 1S'P a ex 1SP b ey and 1S'P b ey
Sum of focal radii
SP S 'P
2a 2b
2
xe
a
.1)e1(a
y
a
x22
2
2
2
2 2
2 2
x y1,
a b
2
2
a
b1
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Distance between foci 2ae 2be
Distance between
directrices
2a/e 2b/e
Tangents at the vertices x=a, x = a y= b, y=b
4. POSITION OF A POINT WITH RESPECT TO AN ELLIPSE
The point P(x1, y1) lies outside, on or inside the ellipse according as
or < 0.
5. CONDITION OF TANGENCY AND POINT OF CONTACT
The condition for the line y = mx + c to be a tangent to the ellipse is that c2=
a2m2+ b2and the coordinates of the points of contact are
Note :
x cos a + y sin a = p is a tangent if p2= a2cos2 + b2sin2 .
lx + my + n = 0 is a tangent if n2= a2l2+ b2m2.
Point form: The equation of the tangent to the ellipse2 2
2 2
x y1
a bat the point 1 1( x , y ) is
1 1
2 2
xx yy1
a b
Slope form: If the line y mx c touches the ellipse2
2 2
x y1
a b, then 2 2 2 2c a m b .
Hence, the straight line 2 2 2y mx a m b always represents the tangents to the
ellipse.
1b
y
a
x2
2
2
2
0,01b
y
a
x2
21
2
21
1b
y
a
x2
2
2
2
222
2
222
2
bma
b,
bma
ma
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Points of contact: Line 2 2 2y mx a m b touches the ellipse2
2 2
x y1
a bat
2 2
2 2 2 2 2 2
a m b,
a m b a m b
Parametric form: The equation of tangent at any point (a cos ,bsin ) is
x ycos sin 1
a b
6. Some Standard Result
The straight line lx my n 0 touches the ellipse2 2
2 2
x y1
a b, if 2 2 2 2 2a l b m m .
The line x cos y sin p touches the ellipse2 2
2 2
x y1
a b, if 2 2 2 2 2a cos b sin p
and that point of contact is2 2a cos b sin
,p p
.
Two tangents can be drawn from a point to an ellipse. The two tangents are real and
distinct or coincident or imaginary according as the given point lies outside, on or inside
the ellipse.
The tangents at the extremities of latus-rectum of an ellipse intersect.
Hyperbola
1. DEFINITION
A hyperbola is the locus of a point which moves in a plane so that the ratio of its
distances from a fixed point (called focus) and a fixed line (called directrix) is a constant
which is greater than one. This ratio is called eccentricity and is denoted by e. For a
hyperbola e > 1.
Let S be the focus, QN be the directrix and P be any point on the hyperbola. Then, by
definition
or PS = e PN, e > 1,
where PN is the length of the perpendicular from P on the directrix QN.
PN
PS
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An Alternate Definition
A hyperbola is the locus of a point which moves in such a way that the difference of its
distances from two fixed points (called foci) is constant.
2. EQUATION OF HYPERBOLA IN STANDARD FORM
The general form of standard hyperbola is
where a and b are constants.
Symmetry
Since only even powers of x and y occur in the above equation, so the curve is
symmetrical about both the axes.
3. TERMS RELATED TO A HYPERBOLA
A sketch of the locus of a moving point satisfying the equation , has been
shown in the figure given above.
Foci If S and S' are the two foci of the hyperbola and their coordinatesd are (ae, 0) and (ae,
0) respectively, then distance between foci is given by SS' = 2ae.
1b
y
a
x2
2
2
2
M'M
B
Axis
Directrix
Conju
gateC
x = a/e
Directrix
A(a,0)
Z
LP(x, y)
Rectum
S(ae, 0)
Latus
x=-a/e
A'
(-a,
0)
S'(-ae, 0)
X'
N
N'
Z'
Y'
B'
L'
X
1b
y
a
x2
2
2
2
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Directries ZM and Z' M' are the two directrices of the hyperbola and their equations are x =
and
x = respectively, then the distance directrices is given by zz' = .
Axes The lines AA' and BB' are called the transverse axis and conjugate axis respectively of
the hyperbola.
The length of transverse axis = AA' = 2a
The length of conjugate axis = BB' = 2b
Centre The point of intersection C of the axes of hyperbola is called the centre of the
hyperbola. All chords, passing through C, are bisected at C.
Vertices The points A (a, 0) and A' (a, 0) where the curve meets the line joining the foci
S and S', are called the vertices of the hyperbola.
Focal Chord A chord of the hyperbola passing through its focus is called a focal chord.
Focal Distances of a Point
The difference of the focal distances of any point on the hyperbola is constant and equal
to the length of the transverse axis of the hyperbola. If P is any point on the hyperbola,
then
S'PSP = 2a = Transverse axis.
e
a
e
a
e
a2
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Latus Rectum If LL' and NN' are the latus rectum of the hyperbola then these lines are
perpendicular to the transverse axis AA', passing through the foci S and S' respectively.
, , , .
Length of latus rectum = LL' = = NN'.
Eccentricity of the HyperbolaWe know that
SP = e PM or SP2= e2PM2or (xae)2+ (y0)2= e2
(xae)2+ y2= (exa)2 x2+ a2e22aex + y2= e2x22aex + a2
x2(e21)y2= a2(e21) .
On comparing with , we get b2= a2(e21) or e =
5. 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.
The conjugate hyperbola of the hyperbola.
is
a
b,aeL
2
a
b,ae'L
2
a
b,aeN
2
a
b,ae'N
2
a
b2 2
2
e
ax'N
1)1e(a
y
a
x22
2
2
2
1b
y
a
x2
2
2
2
2
2
a
b1
1b
y
a
x.,e.i1
b
y
a
x2
2
2
2
2
2
2
2
1
b
y
a
x2
2
2
2
Y
S(0, be)
B(0, b) y
Z
C
B(0, -b) y
S'(0,-
be)
X'
Y'
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Hyperbola
Fundamentals
2 2
2 2 1
x y
a b
2 2
2 2 1
x y
a bor
2 2
2 12
x y
a b
Centre (0, 0) (0, 0)
Length of transverse
axis
2a 2b
Length of conjugate
axis
2b 2a
Foci ( ae,0) (0, be)
Equation of directrices x a /e y b/e
Eccentricity 2 2
2
a be
a
2 2
2
a be
b
Length of latus rectum 22b
a
22a
b
Parametric co-
ordinates
(asec ,btan ) , 0 2 (b sec ,a tan ),0 2
Focal radii 1SP ex a &
1S P ex a
1SP ey b & 1S P ey b
Difference of focal radii
(S P SP )
2a 2b
Tangents at the
vertices
x a, x a y b,y b
Equation of the
transverse axis
y 0 x 0
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Equation of the conjugate
axis
x 0 y 0
6. POSITION OF A POINT WITH RESPECT TO A HYPERBOLA
The point P(x1, y1) lies outside, on or inside the hyperbola according as
= 0 or < 0.
7. CONDITION FOR TRANGENCY AND POINTS OF CONTACT
The condition for the line y = mx + c to be a tangent to the hyperbola is that c2
= a2m2b2and the coordinates of the points of contact are
8. EQUATION OF TANGENT IN DIFFERENT FORMS
Point Form
The equation of the tangent to the hyperbola at the point (x1, y1) is
.
Note :The equation of tangent at (x1, y1) can also be obtained by replacing x2by xx1,
y2by yy1, x by ,y by and xy by . This method is used only when the
equation of hyperbola is a polynomial of second degree in x and y.
Parametric Form The eqn of the tangent to the hyperbola at the point (a
sec, b tan) is
1b
y
a
x2
2
2
2
.01b
y
a
x2
21
2
21
1
b
y
a
x2
2
2
2
222
2
222
2
bma
b,
bma
ma
1b
y
a
x2
2
2
2
1b
yy
a
xx2
1
2
1
2
xx 1
2
yy 1 1 1xy x y
2
1b
y
a
x2
2
2
2
1tanb
ysec
a
x
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Slope FormThe equation of tangent to the hyperbola in terms of slope 'm' is
y = mx
1b
y
a
x2
2
2
2
222 bma