Find the separate equation of the lines represented by the following equations :
(1) `3x^(2) - 10 xy - 8y^(2) = 0`
(2) `x^(2) + 2xy - y^(2) = 0`.

Explanation

(1) `3x^(2) -10xy -8y^(2) =0`
`therefore 3x^(2) - 12xy + 2xy - 8y^(2) = 0`
`therefore 3x(x-4y) +2y(x-4y) = 0`
`therefore (x-4y)(3x +2y) = 0`
`therefore` the separate equations of the lines are `x - 4y = 0 and 3x + 2y = 0`.
(2) `x^(2) + 2xy - y^(2) = 0`
The auxiliary equation is `-m^(2) + 2m + 1 = 0`
`therefore m^(2) - 2m -1 = 0`
`therefore m = (2pmsqrt((-2)^(2)-4(1)(-1)))/(2xx1)`
` = (2 pm sqrt(4+4))/(2)`
` = (2pm 2sqrt(2))/(2) = 1 pm sqrt(2)`
`therefore m_(1) = 1 + sqrt(2) and m_(2) = 1-sqrt(2)` are the slopes of lines.
`therefore` their separate equations are `y = m_(1)x and y = m_(2)x`
i.e., `y = (1 + sqrt(2))x and y = (1-sqrt(2))x.`