Differentiation of parametric equations 317
Determine the equation of the tangent drawn to the
parabola x = 2t 2 , y = 4t at the point t .
At point t , x 1 = 2t
2 , hence
dx 1
dt
= 4t
and
y 1 = 4t , hence
d y 1
dt
= 4
From equation (1),
d y
dx
=
d y
dt
dx
dt
=
4
4t
=
1
t
Hence, the equation of the tangent is:
y − 4t =
1
t
x − 2t 2
Problem 4. The parametric equations of a cycloid
are x = 4(θ − sin θ), y = 4(1 − cosθ). Determine
(a)
d y
dx
(b)
d 2 y
dx 2
(a) x = 4(θ − sin θ),
hence
dx
dθ
= 4 −4 cos θ = 4(1 − cos θ)
y = 4(1 − cos θ), hence
d y
dθ
= 4 sinθ
From equation (1),
d y
dx
=
d y
dθ
dx
dθ
=
4 sinθ
4(1 − cos θ)
=
sin θ
(1 − cos θ)
(b) From equation (2),
d 2 y
dx 2 =
d
dθ
d y
dx
dx
dθ
=
d
dθ
sin θ
1 − cos θ
4(1 − cos θ)
=
(1 − cos θ)(cos θ) − (sin θ)(sin θ)
(1 − cos θ)
2
4(1 − cos θ)
=
cos θ − cos 2 θ − sin 2 θ
4(1 − cos θ) 3
=
cos θ −
cos 2 θ + sin 2 θ
4(1 − cos θ ) 3
=
cos θ − 1
4(1 − cos θ ) 3
=
−(1 − cos θ)
4(1 − cos θ ) 3 =
−1
4(1 − cos θ) 2
Now try the following exercise
Exercise 126 Further problems on
differentiation of parametric equations
1. Given x = 3t − 1 and y = t (t − 1), determine
d y
dx
in terms of t .
1
3
(2t − 1)
2. A parabola has parametric equations: x = t 2 ,
y = 2t . Evaluate
d y
dx
when t = 0.5.
[2]
3. The parametric equations for an ellipse
are x = 4 cos θ, y = sin θ. Determine (a)
d y
dx
(b)
d 2 y
dx 2 .
(a) −
1
4
cot θ (b) −
1
16
cosec 3 θ
4. Evaluate
d y
dx
at θ =
π
6
radians for the
hyperbola whose parametric equations are
x = 3 secθ, y = 6 tanθ.
[ 4 ]
5. The parametric equations for a rectangular
hyperbola are x = 2t , y =
2
t
. Evaluate
d y
dx
when t = 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 =
d y 1
dx 1
(x − x 1 )
Use this in Problems 6 and 7.
6. Determine the equation of the tangent drawn
to the ellipse x = 3 cos θ, y = 2 sinθ at θ =
π
6
.
[y =−1.155x + 4]
7. Determine the equation of the tangent drawn
to the rectangular hyperbola x = 5t , y =
5
t
at
t = 2.
y =−
1
4
x + 5