316 DIFFERENTIAL CALCULUS(a)x=4(θ−sinθ),
hencedx
dθ= 4 −4 cosθ=4(1−cosθ)y=4(1−cosθ), hencedy
dθ=4 sinθFrom equation (1),dy
dx=dy
dθ
dx
dθ=4 sinθ
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)^2Now try the following exercise.Exercise 128 Further problems on differen-
tiation of parametric equations- Givenx= 3 t−1 andy=t(t−1), determine
dy
dx
in terms oft.[
1
3(2t−1)]- A parabola has parametric equations:
x=t^2 , y= 2 t. Evaluatedy
dxwhent= 0. 5
[2]- The parametric equations for an ellipse
arex=4 cosθ,y=sinθ. Determine (a)dy
dx(b)d^2 y
dx^2[
(a)−1
4cotθ (b)−1
16cosec^3 θ]- Evaluate
dy
dxat θ=π
6radians for the
hyperbola whose parametric equations are
x=3 secθ,y=6 tanθ. [4]- 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.- Determine the equation of the tangent drawn
to the ellipsex=3 cosθ,y=2 sinθatθ=
π
6.
[y=− 1. 155 x+4]- Determine the equation of the tangent drawn
to the rectangular hyperbolax= 5 t,y=5
tat
t=2. [y=−1
4x+ 5]29.4 Further worked problems on
differentiation of parametric
equationsProblem 5. The equation of the normal drawn
to a curve at point(x 1 ,y 1 )is given by:y−y 1 =−1
dy 1
dx 1(x−x 1 )Determine the equation of the normal drawn to
the astroidx=2 cos^3 θ, y=2 sin^3 θat the point
θ=π
4x=2 cos^3 θ, hencedx
dθ=−6 cos^2 θsinθy=2 sin^3 θ, hencedy
dθ=6 sin^2 θcosθ