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