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Question
Find the separate equation of the line represented by the following equation:
x2 + 2(cosec α)xy + y2 = 0
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Solution
x2 + 2(cosec α)xy + y2 = 0
i.e. y2 + 2(cosec α)xy + x2 = 0
Dividing by x2, we get,
`("y"/"x")^2 + 2"cosec"alpha. ("y"/"x") + 1 = 0`
`therefore "y"/"x" = (-2 "cosec" alpha +- sqrt(4"cosec"^2 alpha - 4 xx 1 xx 1))/(2xx1)`
`= (-2 "cosec" alpha +- 2sqrt("cosec"^2 alpha - 1))/2`
= - cosec α ± cot α
`therefore "y"/"x" = ("cot" alpha - "cosec" alpha)` and
`"y"/"x" = - ("cosec" alpha + "cot" alpha)`
The separate equations of the lines are
(cosec α - cot α) x + y = 0 and (cosec α + cot α) x + y = 0
Notes
Answer in the textbook is incorrect.
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