1515 conics

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Conics

Transcript of 1515 conics

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Conics

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Conic Sections(1) Circle

A circle is formed when

i.e. when the plane is perpendicular to the axis of the cones.

2

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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

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Conic Sections(3) Parabola

A parabola is formed when

i.e. when the plane is parallel to a generator.

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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

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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

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Form the definition of parabola,PF = PN

axyax 22)(222 )()( axyax

22222 22 aaxxyaaxx

axy 42 standard equation of a parabola

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mid-point of FM = the origin (O) = vertex

length of the latus rectum = LL’= 4a

vertex

latus rectum (LL’)

axis of symmetry

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Other forms of Parabola

axy 42

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Other forms of Parabola

ayx 42

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Other forms of Parabola

ayx 42

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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)

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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

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22422222 )( caayaxca

)()( 22222222 caayaxca 222 cabLet

222222 bayaxb

12

2

2

2

by

ax standard equation of

an ellipse

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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

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Other form of Ellipse

12

2

2

2

ay

bx

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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)

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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

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42222222 )( acayaxac

)()( 22222222 acayaxac 222 acbLet

222222 bayaxb

12

2

2

2

by

ax standard equation of

a hyperbola

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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

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asymptote

xaby equation of

asymptote :

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Other form of Hyperbola :

12

2

2

2

bx

ay

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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.

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equation of asymptote : 0yx

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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

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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.

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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

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OR

0)(2

)(2

)(2 111111 FyyExxDyCyyxxyBAx

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Conics Parabola Ellipse HyperbolaGraph

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

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Conics Parabola Ellipse HyperbolaGraph

Standard Equation axy 42 12

2

2

2

by

ax 12

2

2

2

by

ax

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Conics Parabola Ellipse HyperbolaGraph

Directrix x = -a ,eax ,

eax

PNPFe 1 PN

PFe 1

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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)

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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)

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Conics Parabola Ellipse HyperbolaGraph

Length of lantus rectum LL’

4aab22

ab22

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Conics Parabola Ellipse HyperbolaGraph

Asymptotes ---- ----x

aby

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Conics Parabola Ellipse HyperbolaGraph

Parametric representation of P

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