Question

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

Answer 1

It is given that the slope of the first line, m₁ = 2.

Let the slope of the other line be m₂.

The angle between the two lines is 60°.

∴ tan 60° = `("m"_1 -"m"_2)/(1 + "m"_1"m"_2)`

= `sqrt3 = |(2 - m_2)/(1 + m_1m_2)|`

= `sqrt3 = ±((2 - m_2)/(1 + 2m_2))`

=  `sqrt3 = |(2 - m_2)/(1 + m_2)|` =  `sqrt3 = - ((2 - m_2)/(1 + m_2))`

= `sqrt3 (1 + 2m_2) = 2 - m_2` or `sqrt3 (1 + 2m_2) = (2 - m_2)`

= `sqrt3 + 2 sqrt3m_2 + m_2 = 2` or `sqrt3 + 2sqrt3m_2 - m_2 = -2`

= `sqrt3 + (2 sqrt3 + 1) m_2` or `sqrt3 + (2 sqrt3 - 1) = -2`

= ∴ m₂ = `- (2 - sqrt3)/(2 sqrt3 + 1)` or ∴ m₂ = `(-2 + sqrt3)/(2 sqrt3 - 1)`

Case I: m₂ = `- ((2 - sqrt3)/(2 sqrt3 + 1))`

The equation of the line passing through point (2, 3) and having a slope of `(2 - sqrt3)/(2sqrt3 + 1)` is

= `(y - 3) = (2 - sqrt3)/(2 sqrt 3 + 1) (x - 2)`

= `(2 sqrt3 + 1) y - 3 (2 sqrt3 + 1)` = `(2 - sqrt3) x - 2 (2 - sqrt3)`

= `(sqrt(3) - 2)x + (2 sqrt3 + 1)y` = `-4 + 2 sqrt3 + 6 sqrt3 + 3`

= `(sqrt(3) - 2)x + (2 sqrt3 + 1)y` = `-1 + 8 sqrt3`

Case II:  m₂ = `(-2 + sqrt3)/(2 sqrt3 - 1)`

The equation of the line passing through point (2, 3) and having a slope of `-(2 + sqrt3)/(2sqrt3 + 1)` is

= `(y - 3) = (-2 + sqrt3)/(2 sqrt 3 - 1) (x - 2)`

= `(2 sqrt3 - 1) y - 3 (2 sqrt3 - 1)` = `(2 + sqrt3) x + 2 (2 + sqrt3)`

= `(2sqrt(3) - 1)y + (2 + sqrt3)x` = `4 + 2 sqrt3 + 6 sqrt3 - 3`

= `(2 + sqrt(3))x + (2 sqrt3 - 1)y` = `1 + 8 sqrt3`

In this case, the equation of the other line is `(2 + sqrt(3))x + (2 sqrt3 - 1)y` = `1 + 8 sqrt3`.