If the sum of squares of all real values of α , for which the lines 2 x - y + 3 = 0 , 6 x + 3 y + 1 = 0 and…
If the sum of squares of all real values of , for which the lines and do not form a triangle is , then the greatest integer less than or equal to is ________.
Solution
Given lines are $2x - y + 3 = 0$, $6x + 3y + 1 = 0$ and $\alpha x + 2y - 2 = 0$.
These lines will not form a triangle if $\alpha x + 2y - 2 = 0$ is concurrent with $2x - y + 3 = 0$ and $6x + 3y + 1 = 0$ or parallel to either of them.
Case-1: Concurrent lines
$\Rightarrow \begin{vmatrix} 2 & -1 & 3 \\ 6 & 3 & 1 \\ \alpha & 2 & -2 \end{vmatrix} = 0$
$\Rightarrow 2(-6 - 2) + 1(-12 - \alpha) + 3(12 - 3\alpha) = 0$
$\Rightarrow -16 - 12 - \alpha + 36 - 9\alpha = 0$
$\Rightarrow -10\alpha + 8 = 0$
$\Rightarrow \alpha = \frac{4}{5}$
Case-2: Parallel lines
$\frac{-\alpha}{6} = \frac{-2}{3}$, $-\frac{\alpha}{2} = \frac{-2}{-1}$
$\Rightarrow \alpha = 4$, $\alpha = -4$
$\Rightarrow P = 16 + 16 + \frac{16}{25}$
$[P] = [32 + \frac{16}{25}] = 32$