The correct options are
B two real roots if 1 < a < 2
D all real roots if a < – 1
we have (xx+1)2+(xx−1)2=a(a−1)
⇒(xx+1+xx−1)2−2(xx+1)(xx−1)=a(a−1)⇒(2x2x2−1)2−2x2x2−1=a(a−1)
⇒z2−z−a(a−1)=0, where z=2x2x2−1⇒z=a or 1−a
When, z=a, 2x2x2−1=a⇒2x2=ax2−a⇒x=±√aa−2
When, z=1−a, 2x2x2−1=1−a⇒2x2=(1−a)x2−1+a⇒x=±√a−1a+1
∴ x=±√aa−2,±√a−1a+1
If a < – 1 ⇒ All roots are real.
If 1<a<2⇒x=±√a2−ai,±√a−1a+1⇒ Only two roots are real.
If a > 2 ⇒ All roots are real.