Three Dimensional Geometry Question 212
Question: 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
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 the perpendicular and D be the foot of the perpendicular which divides BC in the ratio $ \lambda :1 $ , then $ D( \frac{10\lambda -9}{\lambda +1},\frac{4}{\lambda +1},\frac{-\lambda +5}{\lambda +1} ) $ . …(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( \frac{10\lambda -9}{\lambda +1} )-4( \frac{4}{\lambda +1} )-6( \frac{-\lambda +5}{\lambda +1} )=0 $ Or $ \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} ) $
BETA
