5x^2 – 8xy + 3y^2 = 0

Question

Find the combined equation of a pair of lines passing through the origin and perpendicular to the lines represented by following equation :

5x² – 8xy + 3y² = 0

Answer 1

Comparing the equation 5x² – 8xy + 3y² = 0

with ax² + 2hxy + by² = 0, we get

a = 5, 2h = -8, b = 3

Let m₁ and m₂ be the slopes of the lines represented by 5x² – 8xy + 3y² = 0.

∴ m₁ + m₂ = \(\cfrac{-2h}{b}\) = \(\cfrac{8}{3}\)

and m₁ m₂ = a/b = 5/3 …(1)

Now required lines are perpendicular to these lines

∴ their slopes are -1 /m₁ and -1/m₂ Since these lines are passing through the origin, their separate equations are

∴ their combined equation is

(x + m₁y) (x + m₂y) = 0