The value of a for which both the roots of the equation `(1-a^(2))x^(2)+2ax-1=0` lie between 0 and 1, will always be greater than

Answer

To find the value of \( a \) for which both roots of the equation

\[ (1 - a^2)x^2 + 2ax - 1 = 0 \]

lie between 0 and 1, we will follow these steps:

### Step 1: Identify the nature of the quadratic equation
The given equation is a quadratic equation in the form \( Ax^2 + Bx + C = 0 \), where:
- \( A = 1 - a^2 \)
- \( B = 2a \)
- \( C = -1 \)

For the roots to be real, the discriminant must be non-negative. The discriminant \( D \) is given by:

\[ D = B^2 - 4AC = (2a)^2 - 4(1 - a^2)(-1) \]

### Step 2: Calculate the discriminant
Substituting the values of \( A \), \( B \), and \( C \):

\[ D = 4a^2 + 4(1 - a^2) = 4a^2 + 4 - 4a^2 = 4 \]

Since \( D = 4 \) is always positive, the roots are always real.

### Step 3: Determine the conditions for the roots to lie between 0 and 1
For both roots \( r_1 \) and \( r_2 \) to lie between 0 and 1, we need to satisfy the following conditions:

1. \( f(0) > 0 \)
2. \( f(1) < 0 \)

### Step 4: Evaluate \( f(0) \)
Substituting \( x = 0 \):

\[ f(0) = (1 - a^2)(0)^2 + 2a(0) - 1 = -1 \]

This condition does not help us since \( f(0) \) is always negative.

### Step 5: Evaluate \( f(1) \)
Substituting \( x = 1 \):

\[ f(1) = (1 - a^2)(1)^2 + 2a(1) - 1 = 1 - a^2 + 2a - 1 = -a^2 + 2a \]

We need \( f(1) < 0 \):

\[ -a^2 + 2a < 0 \]

Rearranging gives:

\[ a^2 - 2a > 0 \]

Factoring:

\[ a(a - 2) > 0 \]

### Step 6: Analyze the inequality
The critical points are \( a = 0 \) and \( a = 2 \). The intervals to test are:

1. \( (-\infty, 0) \)
2. \( (0, 2) \)
3. \( (2, \infty) \)

Testing these intervals:
- For \( a < 0 \): \( a(a - 2) > 0 \) is true.
- For \( 0 < a < 2 \): \( a(a - 2) < 0 \) is false.
- For \( a > 2 \): \( a(a - 2) > 0 \) is true.

Thus, the solution to the inequality is:

\[ a < 0 \quad \text{or} \quad a > 2 \]

### Conclusion
Since we are looking for the value of \( a \) such that both roots lie between 0 and 1, we conclude that \( a \) must be greater than 2.

Thus, the value of \( a \) for which both roots of the equation lie between 0 and 1 will always be greater than:

\[ \boxed{2} \]