Conics

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) Parabola

A 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

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

Ellipses

An ellipse is the locus of a variable point on a plane so that the sum of its distance from two fixed points is a constant.

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

by

ax 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 = ab22

Other form of Ellipse

12

2

2

2

ay

bx

Hyperbolas

A hyperbola is the locus of a variable point on a plane such that the difference of its distance from two fixed points is a constant.

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

42222222 )( acayaxac

)()( 22222222 acayaxac 222 acbLet

222222 bayaxb

12

2

2

2

by

ax standard equation of

a hyperbola

vertextransverse axis

conjugate axis

lactus rectum

length of lactus rectum = ab22

length of the semi-transverse axis = a

length of the semi-conjugate axis = b

asymptote

xaby equation of

asymptote :

Other form of Hyperbola :

12

2

2

2

bx

ay

Rectangular Hyperbola

If b = a, then

222 ayx 12

2

2

2

by

ax

12

2

2

2

bx

ay 222 axy

The hyperbola is said to be rectangular hyperbola.

equation of asymptote : 0yx

If the rectangular hyperbola x2 – y2 = a2 is rotated through 45o about the origin, then the coordinate axes will become the asymptotes.

equation becomes :2

2axy

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.

dxdy

dxdy

OR

0)(2

)(2

)(2 111111 FyyExxDyCyyxxyBAx

Conics Parabola Ellipse HyperbolaGraph

Definition PF = PN PF1 + PF2 = 2a | PF1 + PF2 | = 2a

Conics Parabola Ellipse HyperbolaGraph

Standard Equation axy 42 12

2

2

2

by

ax 12

2

2

2

by

ax

Conics Parabola Ellipse HyperbolaGraph

Directrix x = -a ,eax ,

eax

PNPFe 1 PN

PFe 1

Conics Parabola Ellipse HyperbolaGraph

Vertices (0,0) A1(a,0), A2(-a,0), B1(0,b), B2(0,-b)

A1(a,0), A2(-a,0)

Conics Parabola Ellipse HyperbolaGraph

Axes axis of parabola = the x-axis

major axis = A1A2

minor axis =B1B2

transverse axis =A1A2

conjugate axis =B1B2

where B1(0,b), B2(0,-b)

Conics Parabola Ellipse HyperbolaGraph

Length of lantus rectum LL’

4aab22

ab22

Conics Parabola Ellipse HyperbolaGraph

Asymptotes ---- ----x

aby

Conics Parabola Ellipse HyperbolaGraph

Parametric representation of P

)2,( 2 atat )sin,cos( ba )tan,sec( ba