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 $x+y-z=0=x-2 y+3 z-5$ 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 $= \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & -1 \\ 1 & -2 & 3\end{vmatrix} =\hat{i}-4 \hat{j}-3 \hat{k}$
$\therefore$ 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}}$