Question

1/a^2+ax+x^2-1/a^2-ax+x^2+2ax/a^4+a^2x^2+x^4​

Answer 1

Answer:

Let f(x)=(1−a

2

)x

2

+2ax−1. then f(x)=0 has roots between 0 and 1 if

(1) D≥0

(2) (1−a

2

)f(0)>0 and (1−a

2

)f(1)>0

Now, D≥0⇒4a

2

+4(1−a

2

)>0 which is always true.

(1−a

2

)f(0)>0⇒−(1−a

2

)>0⇒a

2

−1>0⇒a<−1 and a>1 ...(i)

and (1−a

2

)f(1)>0⇒(1−a

2

)(2a−a

2

)>0

⇒a(a−1)(a+1)(a−2)>0⇒a<−1 or a>2 or 0<a<1 ...(ii)

From (i) and (ii) we get

a<−1 and a>2