A particle `p` moves from the point `A(0,4)` to the point `10 ,-4)` . The particle `P` can travel the upper-half plane `{(x , y)|ygeq}` at the speed of `1m//s` and the lower-half plane `{(x , y)|ylt=0}` at the speed of 2 m/s. The coordinates of a point on the x-axis, if the sum of the squares of the travel times of the upper- and lower-half planes is minimum, are (1, 0) (b) (2, 0) (c) (4, 0) (d) (5, 0)

Options

A. (1,0)
B. (2,0)
C. (4,0)
D. (5,0)

Answer

B

Explanation

Let the point on the the axis be (c,0). Sum of the square of travel times is

`T=((sqrt(c^2+16))/(1))^2+(sqrt((10-c)^2+16)/(2))^2`
`=c^2+16+(116+c^2-20c)/(4)`
`=(5)/(4)c^2-5c+45=(5)/(4)(c^2-4c+36)`
Therefore, T is minimum at `c=2`.