AP EAMCET Engineering 21 Apr 2019 Shift 1 Paper

Consider the given information,
x=a(t+sin‌t)‌‌‌‌‌⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅(I)
And,
y=a(1−cos‌t)‌‌‌‌‌⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅(II)
Differentiate equation (I) w.r.t t then,
‌

dx

dt

=a(1+cos‌t)
Differentiate equation (II) w.r.t t then,
‌

dy

dx

=a‌sin‌t Then,
‌

dy

dx

=‌

‌ dy dt

dx dt

=‌

a‌sin‌t

a(1+cos‌t)

=‌

sin‌t

(1+cos‌t)

Now
‌

d2y

dx2

=‌

(1+cos‌t)‌cos‌t−sin‌t(−sin‌t)

(1+cos‌t)2

×‌

1

a(1+cos‌t)

‌‌=‌

(cos‌t+1)

a(1+cos‌t)3

‌‌=‌

1

a(1+cos‌t)2

At‌t‌‌=‌

2π

3

‌

d2y

dx2

‌‌=‌

1

a(‌ 1 2 )2

‌‌=‌

4

a