Question

Find the separate equation of the line represented by the following equation:

x² + 2xy tan α - y² = 0

Answer 1

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)