Two lines L 1 -x-5- y 3- - z -2 - L 2 -x- y -1 - z 2- are coplanar- Then- - can take value s

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Skew Lines

Two lines L...

Question

Two lines $L_{1} : x = 5, \frac{y}{3 - α} = \frac{z}{- 2}, L_{2} : x = α, \frac{y}{- 1} = \frac{z}{2 - α}$ are coplanar. Then, $α$ can take value $(s)$

1, 4, 5

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1, 2, 5

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3, 4, 5

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2, 4, 5

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Solution

The correct option is B $1, 4, 5$
The equations of given lines can be written as,
$L_{1} : x = 5, \frac{y}{3 - α} = \frac{z}{- 2}, L_{2} : x = α, \frac{y}{- 1} = \frac{z}{2 - α}$
Since these lines are coplanar.
Therefore, $∣ ∣ ∣ ∣ \begin{matrix} 5 - α & 0 - 0 & 0 - 0 0 & 3 - α & - 2 0 & - 1 & 2 - α \end{matrix} ∣ ∣ ∣ ∣ = 0$
$\Rightarrow (5 - α) (3 - α) (2 - α) - 2 = 0$
$\Rightarrow (5 - α) [6 - 3 α - 2 α + α^{2} - 2] \Rightarrow (5 - α) (α - 1) (α - 4) = 0$
$\Rightarrow α = 1, 4, 5$

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Two lines L1:x=5,y3−α=z−2,L2:x=α,y−1=z2−α are coplanar. Then, α can take value (s)

Two lines $L_{1} : x = 5, \frac{y}{3 - α} = \frac{z}{- 2}, L_{2} : x = α, \frac{y}{- 1} = \frac{z}{2 - α}$ are coplanar. Then, $α$ can take value $(s)$