Question

If the lines \( 3x+y-4=0 \), \( x - \alpha y + 10 = 0 \), \( \beta x + 2y + 4 = 0 \) and \( 3x + y + k = 0 \) represent the sides of a square, then find \( \alpha \beta (k+4)^2 \).

• A. \( -256 \)
• B. \( -512 \)
• C. \( -128 \)
• D. \( -1024 \)

Hint:

To determine a square from four lines, check perpendicularity and distance conditions.

Answer 1

The Correct Option is B

Step 1: Condition for a Square
For four lines to form a square, the slopes of perpendicular lines must satisfy: \[ m_1 \times m_2 = -1 \] Step 2: Finding \( \alpha, \beta, k \)
Using the conditions for perpendicularity, we solve for \( \alpha, \beta, k \) and compute: \[ \alpha \beta (k+4)^2 = -512 \] Thus, the correct answer is \( -512 \).