Solution Given that x = at2, y = 2at
So
dx
dt  = 2at   and   dy
dt = 2a
Therefore
dy
dx  =
2
1
2
dy
a
dt
dx
at
t
dt
=
=
Rationalised 2023-24
 MATHEMATICS
136
Example 33 Find dy
dx
, if x = a (θ + sin θ), y = a (1 – cos θ) Solution We have dx
dθ = a(1 + cos θ), dy
dθ  = a (sin θ)
Therefore
dy
dx  =
sin
tan
(1
cos )
2
dy
a
d
dx
a
d
θ
θ
θ =
=
+
θ
θ
ANote  It may be noted here that dy
dx  is expressed in terms of parameter only
without directly involving the main variables x and y Example 34 Find 
2
2
2
3
3
3
dy, if
x
y
a
dx
+
= Solution Let x = a cos3 θ, y = a sin3 θ