question_answer

The coordinates of the foot of the perpendicular drawn from the origin to the line joining the points (\[-\]9, 4, 5) and (10, 0, \[-\]1) will be

A) (-3, 2, 1)           

B) (1, 2, 2)

C) (4, 5, 3)            

D) none of these

Correct Answer: D

Solution :

[d] Let AD be the perpendicular and D be the foot of the perpendicular which divides BC in the ratio \[\lambda :1\], then \[D\left( \frac{10\lambda -9}{\lambda +1},\frac{4}{\lambda +1},\frac{-\lambda +5}{\lambda +1} \right)\]. ...(i) The direction ratios of AD are \[\frac{10\lambda -9}{\lambda +1},\frac{4}{\lambda +1}\]and \[\frac{-\lambda +5}{\lambda +1}\]  and direction ratios of BC are 19, \[-\]4 and \[-\]6. Since \[AD\bot BC,\]we get \[19\left( \frac{10\lambda -9}{\lambda +1} \right)-4\left( \frac{4}{\lambda +1} \right)-6\left( \frac{-\lambda +5}{\lambda +1} \right)=0\] Or \[\lambda =\frac{31}{28}\] Hence, on putting the value of \[\lambda \] in (i), we get required foot of the perpendicular, i.e., \[\left( \frac{58}{59},\frac{112}{59},\frac{109}{59} \right)\]