If the two lines r-2-2k-k- and r- -k-2-k- intersect at a point- then the value of -

Given : L_1 :r=2î 2k̂+λ(ĵ k̂) and L_2:r= ĵ+k̂+μ(î+2ĵ αk̂) Now, Point on L_1≡(2,λ, 2 λ) and on L_2≡(μ,1+2μ,1 αμ) For intersceting line : 2=μ⋯(i) λ=1+2μ⋯(ii) and ...

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Standard XII

Mathematics

Skew Lines

If the two li...

Question

If the two lines $\to r = 2^i - 2^k + λ (^j -^k)$ and $\to r =^j +^k + μ (^i + 2^j - α^k)$ intersect at a point, then the value of $α =$

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Solution

Given : $L_{1} : \to r = 2^i - 2^k + λ (^j -^k)$ and
$L_{2} : \to r =^j +^k + μ (^i + 2^j - α^k)$
Now, Point on $L_{1} \equiv (2, λ, - 2 - λ)$ and
on $L_{2} \equiv (μ, 1 + 2 μ, 1 - α μ)$
For intersceting line :
$2 = μ \dots (i) λ = 1 + 2 μ \dots (i i)$
and
$- 2 - λ = 1 - α μ \dots (i i i)$
From $(i)$ and $(i i) : μ = 2, λ = 5$
putting in $(i i i) : - 2 - 5 = 1 - 2 α$
$∴ α = 4$

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If the two lines →r=2^i−2^k+λ(^j−^k) and →r=^j+^k+μ(^i+2^j−α^k) intersect at a point, then the value of α=

Given : L1:→r=2^i−2^k+λ(^j−^k) and L2:→r=^j+^k+μ(^i+2^j−α^k) Now, Point on L1≡(2,λ,−2−λ) and on L2≡(μ,1+2μ,1−αμ) For intersceting line : 2=μ⋯(i)λ=1+2μ⋯(ii) and −2−λ=1−αμ⋯(iii) From (i) and (ii):μ=2,λ=5 putting in (iii):−2−5=1−2α ∴α=4

If the two lines $\to r = 2^i - 2^k + λ (^j -^k)$ and $\to r =^j +^k + μ (^i + 2^j - α^k)$ intersect at a point, then the value of $α =$