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 = `(- 2"h")/"b"`  and m₁m₂ = `"a"/"b"`  ....(1)

Now, the required lines are perpendicular to these lines. 

∴ their slopes are `- 1/"m"_1` and `- 1/"m"_2`

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

y = `- 1/"m"_1`x   and  y = `- 1/"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

∴ `"x"^2 (-"2h")/"b" "x" + "a"/"b" "y"^2 = 0`   ...[By(1)]

∴ bx² - 2hxy + ay² = 0