Straight Lines Question 5
Question 5 - 2024 (27 Jan Shift 2)
If the sum of squares of all real values of $\alpha$, for which the lines $2 x-y+3=0,6 x+3 y+1=0$ and $\alpha x+2 y-2=0$ do not form a triangle is $p$, then the greatest integer less than or equal to $p$ is
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Answer (32)
Solution
$2 x-y+3=0$
$6 x+3 y+1=0$
$\alpha x+2 y-2=0$
Will not form a $\Delta$ if $\alpha x+2 y-2=0$ is concurrent with $2 x-y+3=0$ and $6 x+3 y+1=0$ or parallel to either of them so
Case-1: Concurrent lines
$\left|\begin{array}{ccc}2 & -1 & 3 \\ 6 & 3 & 1 \\ \alpha & 2 & -2\end{array}\right|=0 \Rightarrow \alpha=\frac{4}{5}$
Case-2 : Parallel lines
$-\frac{\alpha}{2}=\frac{-6}{3}$ or $-\frac{\alpha}{2}=2$
$\Rightarrow \alpha=4$ or $\alpha=-4$
$P=16+16+\frac{16}{25}$
$[P]=\left[32+\frac{16}{25}\right]=32$
