1.7 PARAMETRIC EQ

8/7/2019 1.7 PARAMETRIC EQ

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PARAMETRIC

REPRESENTATIONS OFCONIC SECTIONS


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IntroductionIntroduction

2) xyi ! 4) 22 ! yxiiFor example:

Equations represent a curve in Cartesian coordinate

)(Ufr!

Usin) !riFor example: Ucos2) !rii

Equations represent a curve in polar equations


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)(tfx!

These are called parametric equations for the curve.

)(tgy!

In this method, the x and y-coordinates of points on

the curve are given separately as functions of an

additional variable t, called the parameter.

IntroductionIntroduction


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Parametric equation of circleParametric equation of circle

222 ryx !

The Cartesian equation of circle

can be replaced by the parametric equation

tay

tax

sin

cos

!

!

where (a cos t, a sin t) is a point on the circle


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In this case the parameter , t , has graphical

significance as can be seen in the diagram below.


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)cos1( Uax

Example 1:

Show that the curve whose parametric equation are

x = a (1 + cos ) and y= asin represent a circle.

Solution:

To convert the parametric equations to Cartesian

form we eliminate t as follows

Usin

ay!

!@ 1cosa

xU

a

yUsin

Continue

(1) (2)


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But1sincos

22! UU

Hence squaring and adding equating (1) & (2) giving

11

22

!

a

y

a

x

ayx !

22

1

This represents a circle with centre (9,0) and radius a.


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Example 2:

Describe and graph the curve represented bythe parametric equations x = cos t and y= sin t0 t 360o

Solution:

Continue

Graph


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y

x0

t=0

t = /2

T=

t=3/2

(cos t, sin t)

t

To identify the curve, weeliminate the parameter.

Since

1sincos22

! UU

and since

x = cos t and y = sin t

for every point (x,y) on the curve

This means that all points on thecurve satisfy the equation

1)(sin)(cos 2222 !! ttyx

122! yx

@radius =1 and centre at origin


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Parametric equation of parabolaParametric equation of parabola

axy 42!

The Cartesian equation of parabola


aty

atx

2

2

!

!

where t R

P (at2, 2at) is a point on the parabola


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In this case the parameter , t , has graphical

significance as can be seen in the diagram below.

y

x

0

y

x

0

(x,y)

y=2at

x=at2


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Example 1:Graph the plane curve given parametrically by

a) x = t2 , y= 2t b) x = t2+ 1 , y= 2t

Solution:y

x0

a) b) y

x0 (1,0)

Standard parabola A similar parabola whereeach x coordinate is

increased by 1 unit


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Example 1:Graph the plane curve given parametrically by x = t+1 and

y= t2-2t, -g < t


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To eliminate the parametert as follows

1xt

Substitute into the y = t2 2t

y = (x-1)2 2(x-1)

y = x2 4x +3

y

x0


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Parametric equation of an ellipseParametric equation of an ellipse

12

2

2

2

!

b

y

a

x

The Cartesian equation of an ellipse


tby

tax

sin

cos

!

!

where (a cos t, b sin t) is a point on the ellipse


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Parametric equation ofellipseParametric equation ofellipse

To eliminate the parameterU as follows

Ucos8!x Usin4!y

8

cosx

!@ U4

siny

!U

Substitute into the Pythagorean identity

18

22

!

y

1sincos22

! UU

11664

22

!yx

y

x0 10

10

-10

-10

The graph is an ellipse


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Parametric equation ofellipseParametric equation ofellipseExample 2:

Find parametric equations for the conic section with thegiven equation 25x2 + 9y2 100x + 54y - 44 = 0

Solution:

125

)3(

9

)2( 22!

yx

By completing the square in x and y we obtain

the standard form

so, centre of ellipse (2, -3), major axis on line x=2

since cos2 U + sin2 U = 1, so

Ucos3

2!

xUsi

4

3!

y

therefore x = 2 +3 cos U

y = -3+

5 sin U

-g


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Parametric equation ofHyperbolaParametric equation ofHyperbola

12

2

2

2

!

b

y

a

x

The Cartesian equation of hyperbola


tby

tax

tan

sec

!

!

where (a sec t, b tan t) is a point on the hyperbola


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Parametric equation of hyperbolaParametric equation of hyperbolaExample 1:

Find parametric equations for the conic section with thegiven equation x2 - 16y2 10x +32y - 7 = 0

Solution:

1)1(16

)5( 22

!

y

x

By completing the square in x and y we obtain

the standard form

so, centre of hyperbola (5, 1) & transverse axis

on line y = 1

since sec2 U - tan2 U = 1, so

Usec4

5!

xUtan1!y

therefore x = 5 + 4 sec U

y = 1+

tan U

-g

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1.7 PARAMETRIC EQ