The square of the distance of the point $P(5,6,7)$ from the line $\frac{x - 2}{2} = \frac{y - 5}{3}

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The square of the distance of the point $P(5,6,7)$ from the line $\frac{x - 2}{2} = \frac{y - 5}{3} = \frac{z - 2}{4}$ is equal to :

Accepted Answer

Sol. [Key Concept: Drop a perpendicular from P to the line ($PQ \cdot$ direction $= 0$) to find the foot Q, then compute ${PQ}^{2}$.]$${\overrightarrow{PQ} \cdot \overrightarrow{A} = 0 }{\Rightarrow (2\lambda - 3)2 + (3\lambda - 1)3 + (4\lambda - 5)4 = 0 }{\Rightarrow \lambda = 1 }$$Point $Q = (4,8,6)$$${PQ}^{2} = 6 $$

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