Q48 / 109 · NTA Abhyas (JEE Main) 2020 · Mathematics · Straight Lines

The sum of all the values of $|λ|$ such that the lines $x + 2 y - 3 = 0, 3 x - y - 1 = 0$ and $2 λ x + y - 2 = 0$ cannot form a triangle is

Correct answer: 2.35

Solution

Case–I:

x + 2 y - 3 = 0

2 λ x + y - 2 = 0

are parallel

\Rightarrow \frac{2 λ}{1} = \frac{1}{2} \Rightarrow |λ| = 0.25

Case–II:

3 x - y - 1 = 0

2 λ x + y - 2

are parallel

\Rightarrow \frac{2 λ}{3} = \frac{1}{- 1} \Rightarrow |λ| = 1.5

Case–III: Lines

x + 2 y - 3 = 0, 3 x - y - 1 = 0

2 λ x + y - 2 = 0

are concurrent

\Rightarrow |\begin{array}{l} 2 λ & 1 & - 2 \\ 1 & 2 & - 3 \\ 3 & - 1 & - 1 \end{array}| = 0 \Rightarrow - 10 λ - 8 + 14 = 0 \Rightarrow |λ| = 0.6

\Rightarrow 0.25 + 1.5 + 0.6 = 2.3 5

Hence, the correct answer is 2.35.