If the shortest distance between the lines x-12=y-23=z-3λ and x-21=y-44=z-55 is 13, then the sum of all possible values of λ is:

a1=(1,2,3),a2=(2,4,5),b→1=2i^+3j^+λk^,b→2=i^+4j^+5k^ shortest distance=a2−a1⋅b1×b2b1×b2 b→1×b→2=i^j^k^23λ145=i15−4λ^+j^(λ−10+k^5 b→1×b→2=(15−4λ)2+(λ−10)2+25 S.D. =|(i^+2j^+2k^)⋅[(15−4λ)i^+(λ−10)j^+5k^]|(15−4λ)2+(λ−10)2+25|15−4λ+2λ−20+10|(15−4λ)2+(λ−10)2+25=133(5−2λ)2=225+16λ2−120λ+λ2+100−20λ+2512λ2+75−60λ=17λ2−140λ+3505λ2−80λ+275=0λ2−16λ+55=0(λ−5)(λ−11)=0⇒λ=5,11The sum of all possible values of λ is: 16

If the shortest distance between the lines x-12=y-23=z-3λ and x-21=y-44=z-55 is 13, then the sum of all possible values of λ is:

a1=(1,2,3),a2=(2,4,5),b→1=2i^+3j^+λk^,b→2=i^+4j^+5k^ shortest distance=a2−a1⋅b1×b2b1×b2 b→1×b→2=i^j^k^23λ145=i15−4λ^+j^(λ−10+k^5 b→1×b→2=(15−4λ)2+(λ−10)2+25 S.D. =|(i^+2j^+2k^)⋅[(15−4λ)i^+(λ−10)j^+5k^]|(15−4λ)2+(λ−10)2+25|15−4λ+2λ−20+10|(15−4λ)2+(λ−10)2+25=133(5−2λ)2=225+16λ2−120λ+λ2+100−20λ+2512λ2+75−60λ=17λ2−140λ+3505λ2−80λ+275=0λ2−16λ+55=0(λ−5)(λ−11)=0⇒λ=5,11The sum of all possible values of λ is: 16

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If the shortest distance between the lines $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{λ}$ and $\frac{x - 2}{1} = \frac{y - 4}{4} = \frac{z - 5}{5}$ is $\frac{1}{\sqrt{3}}$ , then the sum of all possible values of $λ$ is:

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Detailed Solution

$a_{1} = (1, 2, 3), a_{2} = (2, 4, 5), {\vec{b}}_{1} = 2 \hat{i} + 3 \hat{j} + λ \hat{k}, {\vec{b}}_{2} = \hat{i} + 4 \hat{j} + 5 \hat{k} shortest distance = \frac{|(a_{2} - a_{1}) \cdot (b_{1} \times b_{2})|}{|b_{1} \times b_{2}|}$

$\begin{array}{l} {\vec{b}}_{1} \times {\vec{b}}_{2} = |\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & λ \\ 1 & 4 & 5 \end{array}| = \hat{i (15 - 4 λ)} + \hat{j} ((λ - 10) + \hat{k} (5) \end{array}$

$|{\vec{b}}_{1} \times {\vec{b}}_{2}| = \sqrt{(15 - 4 λ)^{2} + (λ - 10)^{2} + 25}$

$S.D. = \frac{| (\hat{i} + 2 \hat{j} + 2 \hat{k}) \cdot [(15 - 4 λ) \hat{i} + (λ - 10) \hat{j} + 5 \hat{k}] |}{\sqrt{(15 - 4 λ)^{2} + (λ - 10)^{2} + 25}}$

$\frac{| 15 - 4 λ + 2 λ - 20 + 10 |}{\sqrt{(15 - 4 λ)^{2} + (λ - 10)^{2} + 25}} = \frac{1}{\sqrt{3}}$

$\begin{array}{l} 3 (5 - 2 λ)^{2} = 225 + 16 λ^{2} - 120 λ + λ^{2} + 100 - 20 λ + 25 \\ 12 λ^{2} + 75 - 60 λ = 17 λ^{2} - 140 λ + 350 \\ 5 λ^{2} - 80 λ + 275 = 0 \\ λ^{2} - 16 λ + 55 = 0 \\ (λ - 5) (λ - 11) = 0 \\ \Rightarrow λ = 5, 11 \end{array}$