The lines r-i-j- 2i-k- and r- 2i-j- - i-j-k- intersect for

The correct option is D no value of λ and μ The given lines intersect, if the shortest distance #160;between the lines is zero. We know that the s ...

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Byju's Answer

Standard XII

Mathematics

Condition for Two Lines to be on the Same Plane

The lines r...

Question

The lines $\to r = \to i - \to j + λ (2 \to i + \to k)$ and $\to r = (2 \to i - \to j) + μ (\to i + \to j - \to k)$ intersect for

λ = 1, μ = 1

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λ = 2, μ = 3

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all values of

λ

and

μ

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no value of

λ

and

μ

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Solution

The given lines intersect, if the shortest distance between the lines is zero.
We know that the shortest distance between the lines $\to r$
$= a_{1} + λ b_{1}$ and $\to r = a_{2} + μ b_{2}$ is
$\frac{| (a_{1} - a_{2}) \cdot (b_{1} \times b_{2}) |}{| b_{1} \times b_{2} |}$ ........(i)
Here ${\to a}_{1} = \to i - \to j, {\to b}_{1} = 2 \to i + \to k$ and
${\to a}_{2} = 2 \to i - \to j, {\to b}_{2} = \to i + \to j - \to k$
Consider $b_{1} \times b_{2} = ∣ ∣ ∣ ∣ \begin{matrix} i & j & k 2 & 0 & 1 1 & 1 & - 1 \end{matrix} ∣ ∣ ∣ ∣$
$= i (- 1) - j (- 2 - 1) + k (2 - 0)$
$∴ b_{1} \times b_{2} = - i + 3 j + 2 k$ ......(ii)
$\Rightarrow | b_{1} \times b_{2} | = \sqrt{(- 1)^{2} + (3)^{2} + 2^{2}}$
$= \sqrt{1 + 9 + 4}$
$∴ | b_{1} \times b_{2} | = \sqrt{14}$ ......(iii)
Now Consider $a_{1} - a_{2} = \to i - \to j - (2 \to i - \to j)$
$= \to i - \to j - 2 \to i + \to j$
$∴ a_{1} - a_{2} = - i$ .......(iv)
Substitute equation(ii), (iii) and (iv) in equation(i), we get
$\frac{| (a_{1} - a_{2}) \cdot (b_{1} \times b_{2}) |}{| b_{1} \times b_{2} |} = \frac{| - i . (- i + 3 j + 2 k) |}{\sqrt{14}}$
$= \frac{1}{\sqrt{14}} \neq 0$
Hence, the given lines do not intersect.
Therefore, the correct option is (D) ( no value of $λ$ and $μ$ )

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The lines →r=→i−→j+λ(2→i+→k) and →r=(2→i−→j)+μ(→i+→j−→k) intersect for

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