JEE MAIN - Mathematics (2024 - 8th April Evening Shift - No. 25)
Let $$\mathrm{P}(\alpha, \beta, \gamma)$$ be the image of the point $$\mathrm{Q}(1,6,4)$$ in the line $$\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$$. Then $$2 \alpha+\beta+\gamma$$ is equal to ________
Answer
11
Explanation
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$$\begin{aligned} & \overrightarrow{Q R} \cdot(\hat{i}+2 \hat{j}+3 \hat{k})=0 \\ & (t-1)+(2 t-5) \times 2+(3 t-2) \times 3=0 \Rightarrow t=\frac{17}{14} \\ & \Rightarrow R \equiv\left(\frac{17}{14}, \frac{48}{14}, \frac{79}{14}\right) \\ & \Rightarrow \frac{\alpha+1}{2}=\frac{17}{14}, \frac{\beta+6}{2}=\frac{48}{14}, \frac{\gamma+4}{2}=\frac{79}{14} \\ & 2 \alpha+\beta+\gamma=\frac{68}{14}-2+\frac{96}{14}-6+\frac{158}{14}-4=11 \end{aligned}$$
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