Question

Find the foot of the perpendicular from the point $(5,7,3)$ to the line $\frac{x-15}{3}=\frac{y-29}{8}=\frac{5-z}{5}$

• A. $(9,13,15)$
• B. $(13,19,27)$
• C. $(18,37,0)$
• D. $(0,-11,30)$

Answer 1

The correct option is A $(9,13,15)$

Given point is $P(5,7,3)$ and line $\frac{x-15}{3}=\frac{y-29}{8}=\frac{z-5}{-5}=k$(say) .......(1)
So any point on this line is given as$Q(3k+15,8k+29,-5k+5)$
Now direction ratios of $PQ$ are, $3k+10,8k+22,-5k+2$
Also $PQ\perp \left(1\right)$
$\Rightarrow 3(3k+10)+8(8k+22)-5(-5k+2)=0$
$\Rightarrow 98k+196=0$

$\Rightarrow k=-2$
Hence, foot of perpendicular drawn on the given line is $(9,13,15)$.