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Question
Find angles between the lines `sqrt3x + y = 1 and x + sqrt3y = 1`.
Solution
First line: `sqrt3"x" + "y" = 1` or `"y" = -sqrt3"x" + 1`
Slope = `-sqrt3 = "m"_1`
Second line: `"x" + sqrt3"y" = 1` or `"y" = -1/sqrt3"x" + 1/sqrt3`
∴ Slope `-1/sqrt3 = "m"_2`
The angle between two lines is θ, then
tanθ = `|("m"_1 - "m"_2)/(1 + "m"_1"m"_2)|`
= `|((-sqrt3) - (-1/sqrt3))/(1 + (-sqrt3) (-1/sqrt3))|`
= `|-sqrt3 + 1/sqrt3|/(1 + 1)`
= `|(-3 + 1)/(2sqrt3)|`
= `2/(2sqrt3)`
= `1/sqrt3`
θ = 30°
The angle between the two given lines is 30° and the other angle is 180° - 30° =150°.
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