Question

If the sum of squares of all real values of \( \alpha \), for which the lines \( 2x - y + 3 = 0 \), \( 6x + 3y + 1 = 0 \) and \( \alpha x + 2y - 2 = 0 \) do not form a triangle \( p \), then the greatest integer less than or equal to \( p \) is ....

Answer 1

Correct Answer: 32

Given:
\(2x - y + 3 = 0, \quad 6x + 3y + 1 = 0, \quad ax + 2y - 2 = 0.\)

To not form a triangle, \(ax + 2y - 2 = 0\) must be concurrent or parallel with the other lines.

Solving for concurrent lines: 
\(\frac{2}{6} = \frac{-1}{3} \implies a = \frac{4}{5}.\)

Similarly, for parallel lines:  
\(a = \pm 4.\)

Calculating \(p\):
\(p = \left(\frac{4}{5}\right)^2 + 4^2 + 4^2 = 32.\)

The Correct answer is: 32