Question

Find the combined equation of the pair of lines passing through the origin and perpendicular to the lines represented by 3x² + 2xy – y² = 0.

Answer 1

Let m₁ and m₂ are slopes of lines represented by 3x² + 2xy – y² = 0.

∴ m₁ + m₂ = `-(2h)/b = (-2)/(-1)` = 2

And m₁m₂ = `a/b = 3/(-1)` = – 3

Now required lines are perpendicular to given lines.

∴ Their slopes are `-1/m_1` and `-1/m_2`

And required lines pass through the origin.

∴ Their equations are y = `-1/m_1x` and y = `-1/m_2x`

∴ m₁y = – x and m₂y = – x

∴ x + m₁y = 0 and x + m₂y = 0

Their combined equation is (x + m₁y)(x + m₂y) = 0

∴ x² + (m₁ + m₂)xy + m₁m₂y² = 0

∴ x² + (2)xy + (–3)y² = 0

∴ x² + 2xy – 3y² = 0