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.