Three Dimensional Geometry Question 200
Question: The co-ordinates of the foot of perpendicular drawn from point $ P(1,,0,,3) $ to the join of points $ A(4,,7,,1) $ and $ B(3,,5,,3) $ is
[RPET 2001]
Options:
A) (5, 7, 1)
B) $ ( \frac{5}{3},\frac{7}{3},\frac{17}{3} ) $
C) $ ( \frac{2}{3},\frac{5}{3},\frac{7}{3} ) $
D) $ ( \frac{5}{3},\frac{2}{3},\frac{7}{3} ) $
Show Answer
Answer:
Correct Answer: B
Solution:
Let D be the foot of perpendicular drawn from $ P(1,,0,,3) $ on the line AB joining (4, 7, 1) and (3, 5, 3).
If D divides AB in ratio $ \lambda :1 $ then $ D=( \frac{3\lambda +4}{\lambda +1},,\frac{5\lambda +7}{\lambda +1},,\frac{3\lambda +1}{\lambda +1} ) $ …..(i)
D.r-s of PD are $ 2\lambda +3,,5\lambda +7,,-2 $
D.r-s of AB are $ -1,,-2,,2 $
$ \because $ $ PD,\bot ,AB $ ; \ $ -(2\lambda +3)-2(5\lambda +7)-4=0 $
Þ $ \lambda =\frac{-7}{4} $
Putting the value of $ \lambda $ in (i), we get the point $ D( \frac{5}{3},,\frac{7}{3},,\frac{17}{3} ) $ .
BETA
