The slope of a line is double of the slope of another line. If tangent of the angle between them is `1/3,` find the slopes of the lines.

Explanation

Let the slope of first line be `m_(1)` and slope of second line be `m_(2)`.
If `theta` is the angle between two lines then
`tan theta=|(m_(2)-m_(1))/(1+m_(1)m_(2))|`
But it is given that the tangent of angle `theta=(1)/(3)`
`implies (1)/(3)=|(m_(2)-m_(1))/(1+m_(1)m_(2))|`.......`(1)`
`m_(1)=2m_(2)`.......`(2)`
Put the value of `m_(1)` from equation `(2)` to equation `(1)`,
`(1)/(3)=|(m_(2)-2m_(2))/(1+2m_(2)xxm_(2))|`
`implies (1)/(3)=(+-m_(2))/(1+2m_(2)^(2))`
`implies 2m_(2)^(2)+-2m_(2)+1=0`
`implies (2m_(2)+-1)(m_(2)+-1)=0`
`implies m_(2)=+-1` or `+-(1)/(2)`
Then if `m_(2)=+-1` then `m_(1)=+-2` and if `m_(2)=+-(1)/(2)` then `m_(1)=+-1`
`:.` Slope of a line `m_(1)=+-2` or `+-1`
and slope of other line `m_(2)=+-1` or `+-(1)/(2)`
Therefore, slopes of the lines are (`+-2`or`+-1`) and (`+-1` or `+-(1)/(2)`).