The number of distinct real values of λ , for which the lines x - 1 1 = y - 2 2 = z + 3 λ 2 and x - 3 1 = y - 2 λ 2 = z - 1 2 , are coplanar is

∵ Lines are coplanar ∴ 3 - 1 2 - 2 1 - - 3 1 2 λ 2 1 λ 2 2 = 0 ⇒ 2 0 4 1 2 λ 2 1 λ 2 2 = 0 ⇒ 2 4 - λ 4 + 4 λ 2 - 2 = 0 ⇒ 4 - λ 4 + 2 λ 2 - 4 = 0 ⇒ λ 2 λ 2 - 2 = 0 ⇒ λ = 0 , 2 , - 2

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Mathematics - Vector Question with Solution | TestHub

MathematicsVectorGeneral definitions, Angle between vectors, Section formula, Geometrical resultsMedium2 minPYQ_2016

MathematicsMediumsingle choice

The number of distinct real values of $λ$ , for which the lines $\frac{x - 1}{1} = \frac{y - 2}{2} = \frac{z + 3}{λ^{2}}$ and $\frac{x - 3}{1} = \frac{y - 2}{λ^{2}} = \frac{z - 1}{2}$ , are coplanar is

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$∵$ Lines are coplanar

$∴$ $|\begin{matrix} 3 - 1 & 2 - 2 & 1 - (- 3) \\ 1 & 2 & λ^{2} \\ 1 & λ^{2} & 2 \end{matrix}| = 0$

$\Rightarrow$ $|\begin{matrix} 2 & 0 & 4 \\ 1 & 2 & λ^{2} \\ 1 & λ^{2} & 2 \end{matrix}| = 0$

$\Rightarrow$ $2 (4 - λ^{4}) + 4 (λ^{2} - 2) = 0$

$\Rightarrow$ $4 - λ^{4} + 2 λ^{2} - 4 = 0$

$\Rightarrow$ $λ^{2} (λ^{2} - 2) = 0$

$\Rightarrow$ $λ = 0, \sqrt{2}, - \sqrt{2}$

Stream:JEESubject:MathematicsTopic:VectorSubtopic:General definitions, Angle between vectors, Section formula, Geometrical results

⏱ 2mℹ️ Source: PYQ_2016

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