Higher Engineering Mathematics, Sixth Edition

Differentiation of parametric equations 317

Determine the equation of the tangent drawn to the

parabolax= 2 t^2 ,y= 4 tat the pointt.

At pointt,x 1 = 2 t^2 , hence

dx 1

dt

= 4 t

and y 1 = 4 t, hence

dy 1

dt

= 4

From equation (1),

dy

dx

=

dy

dt

dx

dt

=

4

4 t

=

1

t

Hence, the equation of the tangent is:

y− 4 t=

1

t

(

x− 2 t^2

)

Problem 4. The parametric equations of a cycloid

arex= 4 (θ−sinθ),y= 4 ( 1 −cosθ). Determine

(a)

dy

dx

(b)

d^2 y

dx^2

(a) x= 4 (θ−sinθ),

hence

dx

dθ

= 4 −4cosθ= 4 ( 1 −cosθ)

y= 4 ( 1 −cosθ), hence

dy

dθ

=4sinθ

From equation (1),

dy

dx

=

dy

dθ

dx

dθ

=

4sinθ

4 ( 1 −cosθ)

=

sinθ

( 1 −cosθ)

(b) From equation (2),

d^2 y

dx^2

=

d

dθ

(

dy

dx

)

dx

dθ

=

d

dθ

(

sinθ

1 −cosθ

)

4 ( 1 −cosθ)

=

( 1 −cosθ)(cosθ)−(sinθ)(sinθ)

( 1 −cosθ)^2

4 ( 1 −cosθ)

=

cosθ−cos^2 θ−sin^2 θ

4 ( 1 −cosθ)^3

=

cosθ−

(

cos^2 θ+sin^2 θ

)

4 ( 1 −cosθ)^3

=

cosθ− 1

4 ( 1 −cosθ)^3

=

−( 1 −cosθ)

4 ( 1 −cosθ)^3

=

− 1

4 ( 1 −cosθ)^2

Now try the following exercise

Exercise 126 Further problemson

differentiation of parametric equations

1. Givenx= 3 t−1andy=t(t− 1 ), determine
   dy
   dx

in terms oft.

[

1

3

( 2 t− 1 )

]

2. A parabola has parametric equations:x=t^2 ,
   y= 2 t.Evaluate

dy

dx

whent= 0 .5. [2]

3. The parametric equations for an ellipse
   arex=4cosθ,y=sinθ. Determine (a)

dy

dx

(b)

d^2 y

dx^2

.

[

(a)−

1

4

cotθ(b)−

1

16

cosec^3 θ

]

4. Evaluate

dy

dx

at θ=

π

6

radians for the

hyperbola whose parametric equations are

x=3secθ,y=6tanθ.[4]

5. The parametric equations for a rectangular
   hyperbola are x= 2 t, y=

2

t

.Evaluate

dy

dx

whent= 0 .40. [−6.25]

The equation of a tangent drawn to a curve at

point(x 1 ,y 1 )is given by:

y−y 1 =

dy 1

dx 1

(x−x 1 )

Use this in Problems 6 and 7.

6. Determine the equation of the tangent drawn
   to the ellipsex=3cosθ,y=2sinθatθ=

π

6

.

[y=− 1. 155 x+4]

7. Determine the equation of the tangent drawn
   to the rectangular hyperbolax= 5 t,y=

5

t

at

t=2. [

y=−

1

4

x+ 5

]