Question

The length of the perpendicular from the point $(2,-1,4)$ on the straight line, $\frac{x+3}{10}=\frac{y-2}{-7}=\frac{z}{1}$ is :

• A. less than $2$
• B. greater than $2$ but less than $3$
• C. greater than $3$ but less than $4$
• D. greater than $4$

Answer 1

The correct option is C greater than $3$ but less than $4$

Given,
$P=(2,-1,4)$ and equation of line $\frac{x+3}{10}=\frac{y-2}{-7}=\frac{z}{1}$

Any point $B$ on the line is$(10\lambda -3,-7\lambda +2,\lambda )$

Now,
Direction ratios of $PB=(10\lambda -5,-7\lambda +3,\lambda -4)$
Direction ratios of the line $=(10,-7,1)$
As the lines are perpendicular, therefore dot product of given lines are $0$.
$\Rightarrow 10(10\lambda -5)-7(-7\lambda +3)+(\lambda -4)=0$
$\Rightarrow 100\lambda -50+49\lambda -21+\lambda -4=0$
$\Rightarrow 150\lambda =75$
$\Rightarrow \lambda =\frac{1}{2}$
Point $B=(2,-\frac{3}{2},\frac{1}{2})$
$|PB|=\frac{5}{\sqrt{2}}=3.53$