Question

Find the image of the point $(5,7,3)$ in 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

Answer: (B)

$(13,19,27)$

The 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$Q(3k+15,8k+29,-5k+5)$
Assume Image of $P$ in the line (1) is $M$, so $PM$ will be perpendicular to the given line and $Q$ will be midpoint of $PM$.
Now direction ratios of $PQ$ are, $3k+10,8k+22,-5k+2$
also $PQ\perp \left(1\right)$
$\Rightarrow 3(3k+10)+8(8k+22)-5(-5k+2)=0$
$\Rightarrow 98k+196=0\Rightarrow k=-2$
Thus $Q=(9,13,15)$
Since $Q$ is mid point of $PM$
Therefore, coordinate of $M$ is $(13,19,27)$
Hence, option 'B' is correct.