The coordinates of the foot of the perpendicular drawn from the origin to the line joining the point `(-9,4,5)a n d(10 ,0,-1)` will be a. `(-3,2,1)` b. `(1,2,2)` c. `(4,5,3)` d. none of these

Options

A. (-3,2,1)
B. (1,2,2,)
C. (4,5,3)
D. none of these

Answer

D

Explanation

Let AD be the perpendicular and D be the foot of the perpendicular which divides BC in the ratio `lamda:1`, then
`D((10lamda-9)/(lamda+1),(4)/(lamda+1),(-lamda+5)/(lamda+1))" "(i)`

The direction ratio of AD are `(10lamda-0)/(lamda+1),(4)/(lamda+1)and(-lamda+5)/(lamda+1)` and direction ratios of BC are 19,-4 and -6.
Since `ADbotBC`, we get
`19((10lamda-9)/(lamda+1))-4((4)/(lamda+1))-6((-lamda+5)/(lamda+1))=0`
or `lamda=(31)/(28)`
Hence, on putting the value of `lamda` in (i) ,we get required foot of the perpendicular, i.e., `((58)/(59),(112)/(59),(109)/(59))`.