If the sum of squares of all real values of , for which the lines 2x – y + 3 = 0, 6x + 3y + 1 = 0 and do not form a triangle is p, then the greatest integer less than or equal to p is __________. [2024]

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Solution

Q.
If the sum of squares of all real values of $α$ , for which the lines 2x – y + 3 = 0, 6x + 3y + 1 = 0 and $α x + 2 y - 2 = 0$ do not form a triangle is p, then the greatest integer less than or equal to p is __________. [2024]

Ans.
(32)

We have, 2x – y + 3 = 0 ... (i)

6x + 3y + 1 = 0 ... (ii)

$α x + 2 y - 2 = 0$ ... (iii)

Case I. If the lines are concurrent then they do not form a triangle

$∴$ $| \begin{matrix} 2 & - 1 & 3 \\ 6 & 3 & 1 \\ α & 2 & - 2 \end{matrix} |$ $= 0$

$\Rightarrow α (- 1 - 9) - 2 (2 - 18) - 2 (6 + 6) = 0$

$\Rightarrow - 10 α + 32 - 24 = 0 \Rightarrow 10 α = 8 \Rightarrow α = \frac{4}{5}$

Case II. If the lines are parallel then they do not form a triangle.

If the lines (i) and (iii) are parallel,

$∴ \frac{2}{α} = \frac{- 1}{2} \neq \frac{- 3}{2} \Rightarrow α = - 4$

If the lines (ii) and (iii) are parallel,

$∴ \frac{6}{α} = \frac{3}{2} \neq \frac{- 1}{2} \Rightarrow α = 4$

$∴$ Sum of squares of all real values of $α$

$= {(\frac{4}{5})}^{2} + {(- 4)}^{2} + {(4)}^{2} = \frac{16}{25} + 16 + 16 = \frac{16}{25} + 32$

$[P] = [32 + \frac{16}{25}] = 32$ .

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