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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.
Solution
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`.
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