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Question
Find the joint equation of the line passing through the origin and having slopes 1 + `sqrt3` and 1 - `sqrt3`
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Solution
Let l1 and l2 be the two lines. Slopes of l1 is 1 + `sqrt3` and that of l2 is 1 - `sqrt3`
Therefore the equation of a line (l1) passing through the origin and having slope is
y = `(1 + sqrt3)"x"`
∴ `(1 + sqrt3)"x" - "y" = 0` ...(1)
Similarly, the equation of the line (l2) passing through the origin and having slope is
y = `(1 - sqrt3)"x"`
∴ `(1 - sqrt3)"x" - "y" = 0` ...(2)
From (1) and (2) the required combined equation is
`[(1 + sqrt3)"x" - "y"][(1 - sqrt3)"x" - "y"] = 0`
∴ `(1 + sqrt3)"x"[(1 - sqrt3)"x" - "y"] - "y"[(1 - sqrt3)"x" - "y"] = 0`
∴ `(1 - sqrt3)(1 + sqrt3)"x"^2 - (1 + sqrt3)"xy" - (1 - sqrt3)"xy" + "y"^2 = 0`
∴ `((1)^2 - (sqrt3)^2)"x"^2 - [(1 + sqrt3) + (1 - sqrt 3)]"xy" + "y"^2 = 0`
∴ `(1 - 3)"x"^2 - 2"xy" + "y"^2 = 0`
∴ `- 2"x"^2 - "2 xy" + "y"^2 = 0`
∴ 2x2 + 2xy - y2 = 0
This is the required combined equation.
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