Straight Lines
Three Lines Not Forming Triangle — Sum of Squares
nta_pyq_2024_jan
Grade 11
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 is $p$, then the greatest integer less than or equal to $p$ is
Step-by-Step Solution
Key Concept: Lines don't form a triangle if: (a) all three are concurrent, or (b) third line is parallel to one of the first two. Case 1 (concurrent): $\begin{vmatrix}2&-1&3\\6&3&1\\\alpha&2&-2\end{vmatrix}=0\Rightarrow\alpha=4/5$. Case 2 (parallel): $\alpha/2=-1/3\Rightarrow\alpha=-4$ or $\alpha/2=6/3=2\Rightarrow\alpha=4$. $p=(-4)^2+4^2+(4/5)^2=16+16+16/25=32.64$. $[p]=32$.
$\alpha\in\{4/5,-4,4\}$. $p=16+16+16/25=32+16/25$. $[p]=32$.
Correct Answer: 32