Three Dimensional Geometry Question 216
Question: 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
Options:
A) (? 3, 2, 1)
B) (1, 2, 2)
C) (4, 5, 3)
D) None of these
Show Answer
Answer:
Correct Answer: D
Solution:
Let AD be perpendicular and D be foot of perpendicular which divide BC in ratio $ \lambda :1, $ then $ D,( \frac{10\lambda -9}{\lambda +1},\frac{4}{\lambda +1},\frac{-\lambda +5}{\lambda +1} ) $ ?..(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,( \frac{10\lambda -9}{\lambda +1} )-4,( \frac{4}{\lambda +1} )-6,( \frac{-\lambda +5}{\lambda +1} )=0 $
$ \Rightarrow \lambda =\frac{31}{28} $ . Hence on putting the value of $ \lambda $ in (i), we get required foot of the perpendicular i.e. $ ( \frac{58}{59},\frac{112}{59},\frac{109}{59} ) $ . 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 .
BETA
