` If the point (a,a) lies between the lines `|x + y| = 2 , ` then which one of the following is correct ?

Options

A. `|a| lt2`
B. `|a| lt sqrt(2)`
C. `|a| lt 1`
D. `|a| lt 1/sqrt(2) `

Explanation

To solve the problem, we need to determine the conditions under which the point (a, a) lies between the lines defined by the equation |x + y| = 2.

### Step-by-Step Solution:

1. **Understanding the Equation**:
The equation |x + y| = 2 can be split into two linear equations:
- \( x + y = 2 \)
- \( x + y = -2 \)

2. **Substituting the Point**:
Since the point (a, a) lies between these two lines, we will substitute x and y with a:
- For the first line:
\[ a + a = 2 \implies 2a = 2 \implies a = 1 \]
- For the second line:
\[ a + a = -2 \implies 2a = -2 \implies a = -1 \]

3. **Determining the Range**:
The point (a, a) lies between the two lines if:
\[ -2 < x + y < 2 \]
Since \( x + y = 2a \), we can rewrite this as:
\[ -2 < 2a < 2 \]

4. **Dividing the Inequalities**:
Dividing the entire inequality by 2 gives:
\[ -1 < a < 1 \]

5. **Conclusion**:
Therefore, the point (a, a) lies between the lines if:
\[ |a| < 1 \]
This means that the correct relation is:
\[ |a| < 1 \]

### Final Answer:
The correct option is that \( |a| < 1 \).