Search any question & find its solution
Question:
Answered & Verified by Expert
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
Options:
Solution:
1632 Upvotes
Verified Answer
The correct answer is:
$(2,5,7)$
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 $B C \perp A P$
$\begin{aligned}
& \Rightarrow(4 r-3) \cdot 4+(6 r+3) \cdot 6+(8 r-8) \cdot 8=0 \Rightarrow r=\frac{1}{2} \\
& \Rightarrow P(4 r, 6 r+2,8 r+3)=P(2,5,7)
\end{aligned}$
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 $B C \perp A P$
$\begin{aligned}
& \Rightarrow(4 r-3) \cdot 4+(6 r+3) \cdot 6+(8 r-8) \cdot 8=0 \Rightarrow r=\frac{1}{2} \\
& \Rightarrow P(4 r, 6 r+2,8 r+3)=P(2,5,7)
\end{aligned}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.