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Question
Find the combined equation of the pair of lines through the origin and making an equilateral triangle with the line y = 3.
Solution
Let OA and OB be the lines through the origin making an angle of 60° with the line y = 3.
∴ OA and OB make an angle of 60° and 120° with the positive direction of the X-axis.
∴ slope of OA = tan 60° = `sqrt3`
∴ equation of the line OA is
y = `sqrt3`x i.e. `sqrt3`x - y = 0
Slope of OB = tan 120° = tan (180° - 60°)
= - tan 60° = - `sqrt3`
∴ equation of the line OB is
y = - `sqrt3`x, i.e. `sqrt3`x + y = 0
∴ required joint equation of the lines is
`(sqrt3"x" - "y")(sqrt3"x" + "y") = 0`
i.e. 3x2 - y2 = 0
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