If the lines y = 3x + 1 and 2y = x + 3 are equally inclined to the line y = mx + 4, `((1)/(2) lt m lt 3)`, then the values of m are

Options

A. `(1+3sqrt(2))/(7)`
B. `(1-3sqrt(2))/(7)`
C. `(1+-3sqrt(2))/(7)`
D. `(1+-5sqrt(2))/(7)`

Answer

D

Explanation

`m_(1) = 3, m_(2)= (1)/(2)`, and `m_(3) = m`
Let the angle between first and third line is `theta_(1)` and between second and third is `theta_(2)`. Then
`tan theta_(1) =(3-m)/(1+3m)` and `tan theta_(2) = (m-(1)/(2))/(1+(m)/(2))`
But `theta_(1)= theta_(2) rArr (3-m)/(1+3m) = (m-(1)/(2))/(1+(m)/(2))`
`rArr 7m^(2) - 2m - 7 = 0 rArr m = (1+-5sqrt(2))/(7)`.