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(A) The shortest distance $d$ between two lines $\frac{x-x_1}{a_1}=\frac{y-y_1}{b_1}=\frac{z-z_1}{c_1}$ and $\frac{x-x_2}{a_2}=\frac{y-y_2}{b_2}=\frac

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$16$

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(A) The shortest distance $d$ between two lines $\frac{x-x_1}{a_1}=\frac{y-y_1}{b_1}=\frac{z-z_1}{c_1}$ and $\frac{x-x_2}{a_2}=\frac{y-y_2}{b_2}=\frac{z-z_2}{c_2}$ is given by $d = \frac{|(\vec{r_2}-\vec{r_1}) \cdot (\vec{b_1} 	imes \vec{b_2})|}{ |\vec{b_1} 	imes \vec{b_2}| }$.Here,$(x_1, y_1, z_1) = (1, 2, 3)$,$(x_2, y_2, z_2) = (2, 4, 5)$,$\vec{b_1} = 2\hat{i} + 3\hat{j} + \lambda\hat{k}$,and $\vec{b_2} = 1\hat{i} + 4\hat{j} + 5\hat{k}$.$\vec{r_2}-\vec{r_1} = (2-1)\hat{i} + (4-2)\hat{j} + (5-3)\hat{k} = \hat{i} + 2\hat{j} + 2\hat{k}$.$\vec{b_1} 	imes \vec{b_2} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 2 & 3 & \lambda \ 1 & 4 & 5 \end{vmatrix} = \hat{i}(15-4\lambda) - \hat{j}(10-\lambda) + \hat{k}(8-3) = (15-4\lambda)\hat{i} + (\lambda-10)\hat{j} + 5\hat{k}$.The scalar triple product is $(\vec{r_2}-\vec{r_1}) \cdot (\vec{b_1} 	imes \vec{b_2}) = 1(15-4\lambda) + 2(\lambda-10) + 2(5) = 15-4\lambda+2\lambda-20+10 = 5-2\lambda$.The magnitude $|\vec{b_1} 	imes \vec{b_2}| = \sqrt{(15-4\lambda)^2 + (\lambda-10)^2 + 5^2} = \sqrt{225-120\lambda+16\lambda^2 + \lambda^2-20\lambda+100 + 25} = \sqrt{17\lambda^2-140\lambda+350}$.Given $d = \frac{1}{\sqrt{3}}$,we have $\frac{1}{\sqrt{3}} = \frac{|5-2\lambda|}{\sqrt{17\lambda^2-140\lambda+350}}$.Squaring both sides: $\frac{1}{3} = \frac{(5-2\lambda)^2}{17\lambda^2-140\lambda+350} \Rightarrow 17\lambda^2-140\lambda+350 = 3(25-20\lambda+4\lambda^2) = 75-60\lambda+12\lambda^2$.Rearranging: $5\lambda^2-80\lambda+275 = 0 \Rightarrow \lambda^2-16\lambda+55 = 0$.Factoring: $(\lambda-5)(\lambda-11) = 0$,so $\lambda = 5$ or $\lambda = 11$.The sum of possible values of $\lambda$ is $5+11 = 16$.

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

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