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Question
Find the joint equation of the pair of lines through the origin and making an equilateral triangle with the line x = 3.
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Solution
Let OA and OB be the lines through the origin making an angle of 60° with the line x = 3.
∴ OA and OB make an angle of 30° and 150° with the positive direction of X-axis.
∴ Slope of OA = tan 30° = `1/sqrt3`
∴ Equation of the line OA is y = `1/sqrt3x`
∴ `sqrt3y` = x
∴ `x - sqrt3"y" = 0`
Slope of OB = tan 150°
= tan (180° – 30°)
= – tan 30°
= `- 1/sqrt3`
∴ Equation of the line OB is y = `-1/sqrt3x`
∴ `sqrt3y` = – x
∴ x + `sqrt3y` = 0
∴ Required combined equation of the lines is
`(x - sqrt3y)(x + sqrt3y)` = 0
i.e. x2 – 3y2 = 0
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