AP EAMCET Engineering 21 Apr 2019 Shift 1 Paper

The given curve is,
y2(x−a)=x2(x+a)
Apply the log on both sides then,
log(y2(x−a))=log(x2(x+a))
2‌log‌y+log(x−a)=2‌log‌x+log(x+a)
Differentiate both sides w.r.t x

‌

2

y

‌

dy

dx

+‌

1

x−a

=‌

2

x

+‌

1

x+a

‌

2

y

‌

dy

dx

=‌

2

x

+‌

1

x+a

−‌

1

x−a

=‌

2(x+a)(x−a)+x(x−a)−x(x+a)

x(x+a)(x−a)

=‌

2x2−2a2−2ax

x(x−a)(x+a)

Further simplify the above,
‌

dy

dx

=‌

x2−a2−ax

x(x−a)(x+a)

x√‌

x+a

x−a

‌ At ‌‌

dy

dx

=0
Thus,
x2−a2−ax=0
Here, p=a2+4a2>0
So, it has two real roots.