The square of the distance of the image of the point (6, 1, 5) in the line x - 1 / 3 = y / 2 = z - 2 / 4

Question

The square of the distance of the image of the point \((6,1,5)\) in the line \(\frac{x-1}{3}=\frac{y}{2}=\frac{z-2}{4}\), from the origin is _____.

Answer 1

Correct answer is : 62 

Let M\((3\lambda + 1,\ 2\lambda,\ 4\lambda + 2)\)

\(\overrightarrow{\mathrm{AM}} \cdot \overrightarrow{\mathrm{b}}=0\)

\(\Rightarrow \quad 9 \lambda-15+4 \lambda-2+16 \lambda-12=0\)

\(\Rightarrow \quad 29 \lambda=29\)

\(\Rightarrow \quad \lambda=1\)

\(\mathrm{M}(4,2,6), \mathrm{I}=(2,3,7)\)

Required Distance \(=\sqrt{4+9+49}=\sqrt{62}\)