If both roots of the equation ` x^(2) -2ax+a^(2)-1=0` lie between -3 and 4 ,then [a] is/are , where [ ] represents the greatest ineger function

Options

A. 1
B. `-1`
C. 2
D. 0

Answer

To solve the problem, we need to analyze the quadratic equation given by:

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

We want to find the values of \( a \) such that both roots of this equation lie between -3 and 4.

### Step 1: Find the Roots
Using the quadratic formula, the roots of the equation \( ax^2 + bx + c = 0 \) are given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

For our equation, \( a = 1 \), \( b = -2a \), and \( c = a^2 - 1 \). Thus, the roots are:

\[ x = \frac{2a \pm \sqrt{(-2a)^2 - 4 \cdot 1 \cdot (a^2 - 1)}}{2 \cdot 1} \]

Calculating the discriminant:

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

So the roots simplify to:

\[ x = \frac{2a \pm 2}{2} = a \pm 1 \]

Thus, the roots are:

\[ x_1 = a + 1 \quad \text{and} \quad x_2 = a - 1 \]

### Step 2: Set the Conditions for the Roots
We need both roots \( x_1 \) and \( x_2 \) to lie between -3 and 4. This gives us two inequalities to solve:

1. \( -3 < a - 1 < 4 \)
2. \( -3 < a + 1 < 4 \)

### Step 3: Solve the Inequalities

**For the first inequality:**

\[ -3 < a - 1 < 4 \]

Breaking it down:

- From \( -3 < a - 1 \):

\[ a > -2 \]

- From \( a - 1 < 4 \):

\[ a < 5 \]

Thus, the first inequality gives us:

\[ -2 < a < 5 \]

**For the second inequality:**

\[ -3 < a + 1 < 4 \]

Breaking it down:

- From \( -3 < a + 1 \):

\[ a > -4 \]

- From \( a + 1 < 4 \):

\[ a < 3 \]

Thus, the second inequality gives us:

\[ -4 < a < 3 \]

### Step 4: Combine the Results
Now we combine the results from both inequalities:

1. From the first inequality: \( -2 < a < 5 \)
2. From the second inequality: \( -4 < a < 3 \)

The overlapping region is:

\[ -2 < a < 3 \]

### Step 5: Determine the Greatest Integer Function Values
The values of \( a \) that satisfy \( -2 < a < 3 \) are:

- The greatest integer less than \( -2 \) is \( -2 \).
- The greatest integer less than \( 3 \) is \( 2 \).

Thus, the possible integer values for \( [a] \) (the greatest integer function of \( a \)) are:

\[ -2, -1, 0, 1, 2 \]

### Final Answer
The values of \( [a] \) are:

\[ [-2, -1, 0, 1, 2] \]