Conic Sections
Conic Sections(1) Circle
A circle is formed when
i.e. when the plane is perpendicular to the axis of the cones.
2
Conic Sections(2) Ellipse
An ellipse is formed when
i.e. when the plane cuts only one of the cones, but is neither perpendicular to the axis nor parallel to the a generator.
2
Conic Sections
(3) ParabolaA parabola is formed when
i.e. when the plane is parallel to a generator.
Conic Sections(4) Hyperbola
A hyperbola is formed when
i.e. when the plane cuts both the cones, but does not pass through the common vertex.
0
A circle is the locus of a variable point on a plane so that its distance (the radius)remains constant from a fixed point (the centre).y
xO
P(x,y)
The standard equation of circle:
where is the centre of the circle and r is its radius.
× The parametric equation of a circle:
× The general equation of a circle:
where is the centre of the circle and
is its radius
2 2 2( ) ( )x h y k r ( , )h k
cos , sinx r y r
2 2 2 2 0x y gx fy c
( , )g f
2 2g f c
ParabolaA parabola is the locus of a variable point on a plane so that its distance from a fixed point (the focus) is equal to its distance from a fixed line (the directrix x = - a).
focus F(a,0)
P(x,y)
M(-a,0) x
y
O
Form the definition of parabola,
PF = PN
axyax 22)(222 )()( axyax
22222 22 aaxxyaaxx
axy 42 standard equation of a parabola
mid-point of FM = the origin (O) = vertex
length of the latus rectum =LL`= 4a
vertex
latus rectum (LL’)
axis of symmetry
Other forms of Parabola
axy 42
Other forms of Parabola
ayx 42
Other forms of Parabola
ayx 42
12.1 Equations of a ParabolaA parabola is the locus of a variable point P which moves in a plane so that its distance from a fixed point F in the plane equals its distance from a fixed line l in the plane.
The fixed point F is called the focus and the fixed line l is called the directrix.
12.1 Equations of a Parabola
The equation of a parabola with focus F(a,0) and directrix x + a =0, where a >0, is y2 = 4ax.
12.1 Equations of a Parabola
X`X is the axis.
O is the vertex.
F is the focus.
MN is the focal chord.
HK is the latus rectum.
The standard equation of parabola:
where is the focus and is the vertex of parabola.
× The parametric equation of a parabola:
× The general equation of a parabola:
with either a=0 or b=0 but both not zero at the same time.
2( ) 4 ( )y k a x h ( ,0)F a
2 , 2x at y at
2 2 2 2 0ax by gx fy c
( , )h k
12.4 Equations of an EllipseAn ellipse is a curve which is the locus of a variable point which moves in a plane so that the sum of its distance from two fixed points remains a constant. The two fixed points are called foci.
P’(x,y)
P’’(x,y)
Let PF1+PF2 = 2a where a > 0
aycxycx 2)()( 2222 2222 )(2)( ycxaycx
222222 )()(44)( ycxycxaaycx
222 44)(4 acxycxa 42222222 2)2( acxaxcycxcxa
42222222222 22 acxaxcyacaxcaxa
22422222 )( caayaxca
)()( 22222222 caayaxca 222 cabLet
222222 bayaxb
12
2
2
2
b
y
a
x standard equation of an ellipse
vertex
major axis = 2a
minor axis = 2b
lactus rectum
length of semi-major axis = a
length of the semi-minor axis = b
length of lactus rectum = a
b22
12.4 Equations of an Ellipse
12.4 Equations of an Ellipse
AB major axis
CD minor axis
A, B, C and D vertices
O centre
PQ focal chord
F focus
RS, R’S’ latus rectum
12.4 Equations of an Ellipse
Other form of Ellipse
12
2
2
2
a
y
b
x
where a2 – b2 = c2
and a > b > 0
12.4 Equations of an Ellipse
12.4 Equations of an Ellipse
. is axis
minor -semi theof that and is axismajor -semi
theoflength theaxis,- on the lie foci itsthen
,0 where,1 ellipsean Given (2)
. is axisminor -semi theof
that and is axismajor -semi theoflength the
,0 where,1 ellipsean Given (1)
e,Furthermor
2
2
2
2
2
2
2
2
b
a
y
baa
y
b
x
b
a
bab
y
a
x
12.4 Equations of an Ellipse
axes. coordinate the toparallel
are axes whoseand )(at is centre whoseellipse
an represent ,1)()(
equation The (3)2
2
2
2
kh,b
ky
a
hx
y
xO
(h, k)
1)()(
2
2
2
2
b
ky
a
hx
The standard equation of ellipse:
where are the foci of the ellipse.
× The parametric equation of an ellipse:
2 22 2 2
2 21,
x ya b and c a b
a b
( ,0)F c
cos , sinx a y b
A hyperbola is a curve which is the locus of a variable point which moves in a plane so that the difference of its distance from two points remains a constant. The two fixed points are called foci.
P’(x,y)
12.7 Equations of a Hyperbola
Let |PF1-PF2| = 2a where a > 0
aycxycx 2|)()(| 2222 2222 )(2)( ycxaycx
222222 )()(44)( ycxycxaaycx
222 44)(4 acxycxa 42222222 2)2( acxaxcycxcxa
42222222222 22 acxaxcyacaxcaxa
42222222 )( acayaxac
)()( 22222222 acayaxac 222 acbLet
222222 bayaxb
12
2
2
2
b
y
a
x standard equation of a hyperbola
vertextransverse axis
conjugate axis
lactus rectum
length of lactus rectum = a
b22
length of the semi-transverse axis = a
length of the semi-conjugate axis = b
12.7 Equations of a Hyperbola
A1, A2 vertices
A1A2 transverse axis
YY’ conjugate axis
O centre
GH focal chord
CD lactus rectum
asymptote
xa
by equation of
asymptote :
12.7 Equations of a Hyperbola
Other form of Hyperbola :
12
2
2
2
b
x
a
y
12.7 Equations of a Hyperbola
Rectangular Hyperbola
If b = a, then
222 ayx 12
2
2
2
b
y
a
x
12
2
2
2
b
x
a
y 222 axy
The hyperbola is said to be rectangular hyperbola.
equation of asymptote : 0yx
12.7 Equations of a Hyperbola
Properties of a hyperbola :
axis.- the toparallel
axis e transvers),(at centre with hyperbola
a represents1)()(
equation The)1(2
2
2
2
x
kh,b
k-y
a
h-x
axis.- the toparallel
axis e transvers),(at centre with hyperbola
a represents1)()(
-equation The)2(2
2
2
2
y
kh,b
k-y
a
h-x
12.7 Equations of a Hyperbola
Parametric form of a hyperbola :
.parameter a is where
tan
sec
by
ax
.1 hyperbola
on the lies)tan,sec(point the
2
2
2
2
b
y
a
x
ba
12.8 Asymptotes of a Hyperbola
.0
asymptotes twohas constants, positive are
, where,1 hyperbola The2
2
2
2
b
y
a
x
bab
y
a
x
12.8 Asymptotes of a Hyperbola
Properties of asymptotes to a hyperbola :
.0
asymptotes twohas1- hyperbola The)1(2
2
2
2
b
y
a
xb
y
a
x
.0 asymptotes
twohas1)()(
hyperbola The)2(2
2
2
2
b
ky
a
hxb
ky
a
hx
12.8 Asymptotes of a Hyperbola
Properties of asymptotes to a hyperbola :
.0 asymptotes
twohas1)()(
hyperbola The)3(2
2
2
2
b
ky
a
hxb
ky
a
hx
Simple Parametric Equations and Locus Problems
x = f(t)
y = g(t)parametric equations
parameter
Combine the two parametric equations into one equation which is independent of t. Then sketch the locus of the equation.
Equation of Tangents to Conicsgeneral equation of conics :
022 FEyDxCyBxyAx
Steps :
(1) Differentiate the implicit equation to find .
(2) Put the given contact point (x1, y1) into
to find out the slope of tangent at that point.
(3) Find the equation of the tangent at that point.
dx
dy
dx
dy
Case I: If , the equation represents a circle with centre at and radius
Case II: If and both have the same sign, the equation represents the standard equation of an ellipse in XY-coordinate system, where
Case III: If and both have opposite signs, the equation represents the standard equation of hyperbola in XY-coordinate system, where
Case IV: If ,the equation represents the standard equation of parabola in XY- coordinate system, where
2 2 0Ax By Gx Fy C
0A B ( , )
2 2
G F
A A
2 2
2 24 4
G F C
A A A
A B
2 2
G FX x and Y y
A B
A B
2 2( )
G FX x and Y y
A B
0 0A or B
2
2 4
G C GX x and Y y
A F AF
With the understanding that occasional degenerate cases may arise, the quadratic curve is
a parabola, if an ellipse, if a hyperbola, if
2 2 0Ax Bxy Cy Dx Ey F 2 4 0B AC
2 4 0B AC
2 4 0B AC
In both ellipse and hyperbola, the eccentricity is the ratio of the distance between the foci to the distance between the vertices.
Suppose the distance PF of a point P from a fixed point F (the focus)is a constant multiple of its distance from a fixed line (the directrix).i.e. , where e is the constant of proportionality. Then the path traced by P is
(a). a parabola if (b). an ellipse of eccentricity e if (c). a hyperbola of eccentricity e if
.PF e PD
1e 1e
1e
Conics Parabola Ellipse Hyperbola
Graph
Definition
PF = PN PF1 + PF2 = 2a | PF1 - PF2 | = 2a
Conics Parabola Ellipse Hyperbola
Graph
Standard Equation axy 42 1
2
2
2
2
b
y
a
x1
2
2
2
2
b
y
a
x
Conics Parabola Ellipse Hyperbola
Graph
Directrix x = -a,
e
ax ,
e
ax
PN
PFe 1 PN
PFe 1
Conics Parabola Ellipse Hyperbola
Graph
Vertices (0,0) A(-a,0), B(a,0), C(0,b), D(0,-b)
A1(a,0), A2(-a,0)
Conics Parabola Ellipse Hyperbola
Graph
Axes axis of parabola = the x-axis
major axis = AB
minor axis =CD
transverse axis =A1A2
conjugate axis =B1B2
where B1(0,b), B2(0,-b)
Conics Parabola Ellipse Hyperbola
Graph
Length of lantus
rectum LL`
4aa
b22
a
b22
Conics Parabola Ellipse Hyperbola
Graph
Asymptotes ---- ----x
a
by
Conics Parabola Ellipse Hyperbola
Graph
Parametric representation of P
)2,( 2 atat )sin,cos( ba )tan,sec( ba