Question

Find the separate equations of the lines given by x² + 2xy tan α − y² = 0 

Answer 1

The given equation of the lines is x² + 2xy tan α − y² = 0 

The auxiliary equation of the given combined equation of the lines is m² − 2m tan α − 1 = 0

∴ m = `(2 tan alpha ± sqrt(4tan^2 alpha + 4))/2`

= `(2 tan alpha ± 2sqrt(tan^2 alpha + 1))/2`

∴ m = tan α ± sec α

∴ m₁ = tan α + sec α and m₂ = tan α − sec α

∴ The separate equations of lines are

y = (tan α + sec α)x and y = (tan α − sec α)x

∴ (sec a + tan a)x − y = 0 and (sec a − tan a)x + y = 0