Distance of origin from the line `(1+sqrt3)y+(1-sqrt3)x=10` along the line `y=sqrt3x+k` (1) `2/sqrt5` (2) `5sqrt2+k` (3) `10` (4) 5

Options

A. `(5)/(sqrt(2))`
B. 5sqrt(2)+k
C. 10
D. 5

Answer

D

Explanation

Slope of the line L`-=(1+sqrt(3))y+(1-sqrt(3))x=10` is
`(sqrt(3)-1)/(sqrt(3)+1) ="tan" 15^(@)`.
Slope of line `y=sqrt(3) x+k " is " sqrt(3) = "tan" 60^(@)`.

In the figure, distance of origin from line L measured along the line `y=sqrt(3)x+k` is OP.
From the figure.
`OP=OM "sec" 45^(@)`
`= (|0+0-10|)/(sqrt((sqrt(3)-1)^(2) + (sqrt(3)+1)^(2)))sqrt(2) = (10)/(sqrt(8))sqrt(2) = 5`