Question

Prove that the combined of the pair of lines passing through the origin and perpendicular to the lines ax² + 2hxy + by² = 0 is bx² - 2hxy + ay² = 0.

Answer 1

Let m₁ and m₂ be the slopes of the lines represented by ax² + 2hxy + by² = 0.

∴ m₁ + m₂ = ${\displaystyle \frac{-2\text{h}}{\text{b}}}$  and m₁m₂ = ${\displaystyle \frac{\text{a}}{\text{b}}}$  ....(1)

Now, the required lines are perpendicular to these lines. 

∴ their slopes are ${\displaystyle -\frac{1}{{\text{m}}_1}}$ and ${\displaystyle -\frac{1}{{\text{m}}_2}}$

Since these lines are passing through the origin, their separate equations are

y = ${\displaystyle -\frac{1}{{\text{m}}_1}}$x   and  y = ${\displaystyle -\frac{1}{{\text{m}}_2}}$x

i.e. m₁y = - x and m₂y = - x

i.e. x + m₁y = 0   and  x + m₂y = 0

∴ their combined equation is

(x + m₁y)(x + m₂y) = 0

∴ x² + (m₁ + m₂)xy + m₁m₂y² = 0

∴ ${\displaystyle {\text{x}}^2\frac{-\text{2h}}{\text{b}}\text{x}+\frac{\text{a}}{\text{b}}{\text{y}}^2=0}$   ...[By(1)]

∴ bx² - 2hxy + ay² = 0