If $\mathrm{A}(3,-1,11), \mathrm{B}(0,2,3), \mathrm{C}(4,8,11)$ are three points, then the coordinates of…

If $\mathrm{A}(3,-1,11), \mathrm{B}(0,2,3), \mathrm{C}(4,8,11)$ are three points, then the coordinates of the foot of the perpendicular drawn from the point $\mathrm{A}$ to the line joining the points $\mathrm{B}$ and $C$ is
  1. $(3,5,7)$
  2. $(5,9,6)$
  3. $(2,5,7)$
  4. $(1,2,3)$

Solution

Given $\mathrm{A}(3,-1,11), \mathrm{B}(0,2,3)$ and $\mathrm{C}(4,8,11)$
Now, Equation of $\mathrm{BC}$ is $\begin{aligned} & \frac{x-0}{4}=\frac{y-2}{6}=\frac{2-3}{8}= \\ & \Rightarrow x=4 r, y=6 r+2, z=8 r+3 \end{aligned}$ Any point on $\mathrm{BC}(4 \mathrm{r}, 6 \mathrm{r}+2,8 \mathrm{r}+3)$ DR's of $\mathrm{BC}=(4-0,8-2,11-3)=(4,6,8)$ DR's of $A P=(4 r-3,6 r+3,8 r-8)$ Since $B C \perp A P$ $\begin{aligned} & \Rightarrow(4 r-3) \cdot 4+(6 r+3) \cdot 6+(8 r-8) \cdot 8=0 \Rightarrow r=\frac{1}{2} \\ & \Rightarrow P(4 r, 6 r+2,8 r+3)=P(2,5,7) \end{aligned}$

Asked in: AP EAMCET 2023 (15 May Shift 2)

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