question_answer

The co-ordinates of the foot of 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 :

                Let AD be perpendicular and D be foot of perpendicular which divide BC in 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 ratio of AD are \[\frac{10\lambda -9}{\lambda +1},\frac{4}{\lambda +1},\frac{-\lambda +5}{\lambda +1}\] and direction ratio of BC are 19, ? 4 and ? 6.                 Since \[AD\,\,\bot \,\,BC\]                 \[\Rightarrow \,\,19\,\left( \frac{10\lambda -9}{\lambda +1} \right)-4\,\left( \frac{4}{\lambda +1} \right)-6\,\left( \frac{-\lambda +5}{\lambda +1} \right)=0\]                 \[\Rightarrow \,\,\lambda =\frac{31}{28}\]. Hence on putting the value of \[\lambda \] in (i), we get required foot of the perpendicular i.e.\[\left( \frac{58}{59},\frac{112}{59},\frac{109}{59} \right)\]. Trick: The line passing through these points is \[\frac{x+9}{19}=\frac{y-4}{-4}=\frac{z-5}{-6}.\] Now co-ordinates of the foot lie on this line, so they must satisfy the given line. But here no point satisfies the line, hence answer is .