Let be the origin, and and be the points on the lines and respectively such that is the shortest distance between the given lines. Then is equal to

9 Explanation: \begin{gather*} \frac{x-5}{4} = \frac{y-4}{1} = \frac{z-5}{3} \\ \frac{x+8}{12} = \frac{y+2}{5} = \frac{z+11}{9} \end{gather*} Let and are two points on given lines. \\ , and \begin{align*} &\vec{MN} \cdot \vec{b}_1 = 0 \\ &4(12\mu - 4\lambda - 13) + 1(5\mu - \lambda - 6) + 3(9\mu - 3\lambda - 16) = 0 \\ \Rightarrow \quad &48\mu - 16\lambda - 52 + 5\mu - \lambda - 6 + 27\mu - 9\lambda - 48 = 0 \\ \Rightarrow \quad &80\mu - 26\lambda = 106 \\ &40\mu - 13\lambda = 53 \tag{i} \end{align*} \begin{align*} &\vec{MN} \cdot \vec{b}_2 = 0 \\ &12(12\mu - 4\lambda - 13) + 5(5\mu - \lambda - 6) + 9(9\mu - 3\lambda - 16) = 0 \\ \Rightarrow \quad &144\mu - 48\lambda - 156 + 25\mu - 5\lambda - 30 + 81\mu - 27\lambda - 144 = 0 \\ \Rightarrow \quad &250\mu - 80\lambda = 330 \\ &25\mu - 8\lambda = 33 \tag{i

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JEE 2024 — Mathematics PYQ

JEE | Mathematics | 2024

Let $O$ be the origin, and $M$ and $N$ be the points on the lines $\frac{x - 5}{4} = \frac{y - 4}{1} = \frac{z - 5}{3}$ and $\frac{x + 8}{12} = \frac{y + 2}{5} = \frac{z + 11}{9}$
respectively such that $M N$ is the shortest distance between the given lines. Then $O M \cdot O N$ is equal to

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