athrm{~s}-2-\mathrm{t}}{2}=\frac{\mathrm{s}-\mathrm{t}+2}{1}=\frac{\mathrm{s}-\mathrm{t}}{2} \\ & \Rightarrow \mathrm{t}=6 \text { and } \mathrm{s}=2 \\ & \Rightarrow \mathrm{P}(6,4,6) \text { and } \mathrm{Q}(2,2,2) \\ & \mathrm{PQ}: \frac{\mathrm{x}-2}{2}=\frac{\mathrm{y}-2}{1}=\frac{\mathrm{z}-2}{2}=\lambda \end{aligned}$$

Let $$\mathrm{F}(2 \lambda+2, \lambda+2,2 \lambda+2)$$

$$\mathrm{A}(1,2,12)$$

$$\overrightarrow{\mathrm{AF}} \cdot \overrightarrow{\mathrm{PQ}}=0$$

$$\therefore \lambda=2$$

So $$F(6,4,6)$$ and $$A F=\sqrt{65}$$

JEE Main 2024 (Online) 29th January Morning Shift Mathematics - 3D Geometry Question 92 English Explanation

19:T543,

माना $$\mathrm{P}(\mathrm{t}, \mathrm{t}-2, \mathrm{t})$$ और $$\mathrm{Q}(2 \mathrm{~s}-2, \mathrm{~s}, \mathrm{~s})$$

PQ की दिक् कोज्याएँ (D.R's) 2, 1, 2 हैं

$$\begin{aligned} & \frac{2 \mathrm{~s}-2-\mathrm{t}}{2}=\frac{\mathrm{s}-\mathrm{t}+2}{1}=\frac{\mathrm{s}-\mathrm{t}}{2} \\ & \Rightarrow \mathrm{t}=6 \text { और } \mathrm{s}=2 \\ & \Rightarrow \mathrm{P}(6,4,6) \text { और } \mathrm{Q}(2,2,2) \\ & \mathrm{PQ}: \frac{\mathrm{x}-2}{2}=\frac{\mathrm{y}-2}{1}=\frac{\mathrm{z}-2}{2}=\lambda \end{aligned}$$

माना $$\mathrm{F}(2 \lambda+2, \lambda+2,2 \lambda+2)$$

$$\mathrm{A}(1,2,12)$$

$$\overrightarrow{\mathrm{AF}} \cdot \overrightarrow{\mathrm{PQ}}=0$$

$$\therefore \lambda=2$$

अतः $$F(6,4,6)$$ और $$AF=\sqrt{65}$$

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