Question

If the point \((a,a)\) is placed in between the lines \(|x+y| = 4\) then

• A. \(|a| = 2\)
• B. \(|a| = 3\)
• C. \(|a|<2\)
• D. \(|a|<3\)

Hint:

For point between parallel lines, the value of the linear expression lies between the constants.

Answer 1

The Correct Option is C

To determine where the point \((a, a)\) is located with respect to the lines represented by the equation \(|x + y| = 4\), we need to analyze these lines and the position of the point.

The given expression \(|x + y| = 4\) denotes two lines:

• \(x + y = 4\)
• \(x + y = -4\)

These lines divide the coordinate plane into different regions. The point \((a, a)\) lies on the line \(y = x\). To find when \((a, a)\) is between these two lines, we substitute \(x = a\) and \(y = a\) into the expressions of the lines:

• For the line \(x + y = 4\):
  \(a + a = 4 \Rightarrow 2a = 4 \Rightarrow a = 2\)
   
• For the line \(x + y = -4\):
  \(a + a = -4 \Rightarrow 2a = -4 \Rightarrow a = -2\)
   

Thus, the point \((a, a)\) will lie between the lines \(\ |x + y| = 4\) if and only if:

• \(-2 < a < 2\)

Expressing this in terms of absolute value gives:

\(|a| < 2\)

Therefore, the correct answer is that \(|a| \lt 2\).

This explains why \(|a| = 2\), \(|a| = 3\), and \(|a| < 3\) are incorrect options, as they don't satisfy the condition for being strictly between the two lines.