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

The sum of the squares of all real values of \( \alpha \) for which the given lines do not form a triangle is calculated by analyzing the conditions for parallelism. The given lines are: 1. \( 2x - y + 3 = 0 \), 2. \( 6x + 3y + 1 = 0 \), and 3. \( \alpha x + 2y - 2 = 0 \). Lines fail to form a triangle if at least two lines are parallel, which occurs when \(\frac{a_1}{a_2} = \frac{b_1}{b_2}\). Considering pairs of lines:

• Lines 1 and 2: \(\frac{2}{6} = \frac{-1}{3}\). Since \(\frac{1}{3} eq \frac{-1}{3}\), lines 1 and 2 are not parallel.
• Lines 1 and 3: \(\frac{2}{\alpha} = \frac{-1}{2}\). Solving for \( \alpha \) yields \( \alpha = -4 \).
• Lines 2 and 3: \(\frac{6}{\alpha} = \frac{3}{2}\). Solving for \( \alpha \) yields \( \alpha = 4 \).

The real values of \( \alpha \) for which the lines do not form a triangle are \( -4 \) and \( 4 \). The sum of the squares of these values is \( (-4)^2 + (4)^2 = 16 + 16 = 32 \). The greatest integer less than or equal to 32 is 32. This computed value of 32 falls within the specified range of [32, 32].