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Question
Find the separate equation of the line represented by the following equation:
x2 + 2xy tan α - y2 = 0
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Solution
The given combined equation is x2 + 2xy tan α - y2 = 0
`x^2/x^2 + (2xy) /x^2 . tan alpha - y^2/x^2`
`1 + (2y)/x . tan alpha - (y/x)^2 = 0`
`(y/x)^2 - (2y)/x . tan alpha - 1 = 0`
put `m = y/x`
`m^2 - 2m tan alpha - 1 = 0`
`m = (- (-2 tan alpha) +- sqrt((-2 tan alpha)^2 - 4 xx 1 xx -1))/(2xx1)`
`m = (2 tan alpha +- sqrt(4 tan^2 alpha + 4))/2`
`m = (2 tan alpha +- sqrt(4 (tan^2 alpha + 1)))/2`
`m = (2 tan alpha +- sqrt(4sec^2alpha))/2`
`m = (2tan alpha +- 2secalpha) /2`
`m = (2(tan alpha +- secalpha))/2`
`m = tanalpha +- sec alpha`
Take (+)
`y/x = (tan alpha - sec alpha)`
`y = (tan alpha + sec alpha)x`
`(tan alpha + sec alpha) x - y = 0` ...(i)
Take (-)
`(tan alpha - sec alpha) x - y = 0` ...(ii)
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