Two lines passes through the point `(3,1)` meet an angle of `60^(@)`. If the slope of one line is `2`, find the equation of second line.

Answer

C

Explanation

Let `m_(1)=2` and slope of second line be `m_(2)`
Now, `tan theta=|(m_(1)-m_(2))/(1+m_(1)m_(2))|`
`implies tan 60^(@)=|(2-m_(2))/(1+2m_(2))|`
`implies sqrt(3)=|(2-m_(2))/(1+2m_(2))|`
`implies +-sqrt(3)=(2-m_(2))/(1+2m_(2))`
Now, `sqrt(3)=(2-m_(2))/(1+2m_(2))`
`implies sqrt(3)+2sqrt(3)m_(2)=2-m_(2)`
`implies m_(2)(2sqrt(3)+1)=2-sqrt(3)`
`implies m_(2)=(2-sqrt(3))/(2sqrt(3)+1)`.......`(1)`
and `-sqrt(3)=(2-m_(2))/(1+2m_(2))`
`implies -sqrt(3)-2sqrt(3)m_(2)=2-m_(2)`
`implies m_(2)(2sqrt(3)-1)=-(sqrt(3)+2)`
`implies m_(2)=(2+sqrt(3))/(1-2sqrt(3))`......`(2)`
Equation of a line passing through the points `(3,1)` and whose slope is `m_(2)` is
`y-1=m_(2)(x-3)`
Put `m_(2)=(2-sqrt(3))/(2sqrt(3)+-1)`
`y-1=(2-sqrt(3))/(2sqrt(3)+1)(x-3)`
`implies y(2sqrt(3)+1)-2sqrt(3)-1=x(2-sqrt(3))-6+3sqrt(3)`
`implies x(2-sqrt(3))-y(2sqrt(3)+1)=5-5sqrt(3)`.
Put `m_(2)=(2+-sqrt(3))/(1-2sqrt(3))`
`y-1=(2+sqrt(3))/(1-2sqrt(3))(x-3)`
`implies y(1-2sqrt(3))-1+2sqrt(3)=x(2+sqrt(3))-6-3sqrt(3)`
`implies x(2+sqrt(3))-y(1-2sqrt(3))=5+5sqrt(3)`.