Given equation is $[1 - a^2]x^2 + 2ax - 1 = 0$. Its discriminant $D = 4$ and roots are $-\frac{1}{a+1}, -\frac{1}{a-1}$. Find the range of $a$ for which this holds.
Step-by-Step Solution
Key Concept: Use the discriminant condition and constraints on root locations (given as specific fractions) to determine the valid parameter range.
From discriminant: $D = 4a^2 + 4(1-a^2) = 4$, which is always satisfied. Given $0 < -\frac{1}{a+1} < 1$ and $-\frac{1}{a-1} < 1$: From $-\frac{1}{a+1} > 0$, we get $a > -1$. From $-\frac{1}{a+1} < 1$, we get $-1 < a$ and $-\frac{1}{a-1} < 1$ gives $a < -1$ or $a > 0$. Combining with $a > 0$ from condition (2), we obtain $a > 2$.
Correct Answer: 2