Two lines passing through the point (2, 3) intersects each other at an angle of `60o`. If slope of one line is 2, find equation of the other line.

Explanation

Slope of first line `m_(1)=2`
Let slope of second line `=m_(2)`
Angle between two lines `=60^(@)`
Now, `(m_(1)-m_(2))/(1+m_(1)m_(2))=+-tan60^(@)`
`implies (2-m_(2))/(1+2m_(2))=+-sqrt(3)`
`implies (2-m_(2))/(1+2m_(2))=sqrt(3)` and `(2-m_(2))/(1+2m_(2))=-sqrt(3)`
`implies sqrt(3)+2sqrt(3)m_(2)=2-m_(2)`
and `sqrt(3)-2sqrt(3)m_(2)=2-m_(2)`
`implies m_(2)(2sqrt(3)+1)=2-sqrt(3)`
and `m_(2)(2sqrt(3)-1)=-(2+sqrt(3))`
`implies m_(2)=(2-sqrt(3))/(2sqrt(3)+1)`
and `m_(2)=(-2(2+sqrt(3)))/(-2sqrt(3)+1)=(2+sqrt(3))/(2sqrt(3)-1)`
Equation of al line passing through the point `(2,3)`
`y-3=m_(2)(x-2)`
`y-3=(2-sqrt(3))/(2sqrt(3)+1)(x-2)`
and `y-3=((2+sqrt(3))/(2sqrt(3)-1)(x-2)`
`implies (2sqrt(3)+1)y-6sqrt(3)-3=(2-sqrt(3))x-4+2sqrt(3)`
and `(2sqrt(3)-1)y-6sqrt(3)+3=-(2+sqrt(3))x-4-2sqrt(3)`
`implies (sqrt(3)-2)x+(2sqrt(3)+1)y=8sqrt(3)-1`
and `(sqrt(3)+2)x+(2sqrt(3)-1)y=8sqrt(3)-7`