Question

The length of the perpendicular from the point $(1,-2,5)$ on the line passing through $(1,2,4)$ and parallel to the line $x+y-z=0=x-2y+3z-5$ is :

• A. $\sqrt{\frac{21}{2}}$
• B. $\sqrt{\frac{9}{2}}$
• C. $\sqrt{\frac{73}{2}}$
• D. 1

Answer 1

Answer: (A)

d.r's of the line $=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 1& -1\\ 1& -2& 3\end{array}\right|=\hat{i}-4\hat{j}-3\hat{k}$
$\therefore$ equation of line is
$\vec{r}=\hat{i}+2\hat{j}+4\hat{k}+\lambda (\hat{i}-4\hat{j}-3\hat{k})$
Let $A(1,2,4)$ and $Pb(1+\lambda ,2-4\lambda ,4-3\lambda )$
$\therefore \overrightarrow{PA}\cdot (\hat{i}-4\hat{j}-3\hat{k})=0$
$\lambda =\frac{1}{2}$
$\Rightarrow P\left(\frac{1}{2},2,\frac{-5}{2}\right)$
$|AP|=\sqrt{\frac{21}{2}}$