Question

Show that the combined equation of pair of lines passing through the origin is a homogeneous equation of degree 2 in x and y. Hence find the combined equation of the lines 2x + 3y = 0 and x − 2y = 0

Answer 1

Let a₁x + b₁y = 0 and a₂x + b₂y = 0 be a pair of lines passing through the origin.

∴ Their combined equation is (a₁x + b₁y)(a₂x + b₂y) = 0

∴ a₁a₂x₂ + a₁b₂xy + b₁a₂xy + b₁b₂y² = 0

∴ (a₁a₂)x² + (a₁b₂ + a₂b₁)xy + (b₁b₂)y² = 0

In this if we put a₁a₂ = a, a₁b₂ + a₂b₁ = 2h, b₁b₂ = b, We get ax² + 2hxy + by² = 0 which is a homogeneous equation of degree 2 in x and y.

Now, on comparing 2x + 3y = 0 and x − 2y = 0 with a₁x + b₁y = 0 and a₂x + b₂y = 0,

we get a₁ = 2, b₁ = 3, a₂ = 1 and b₂ = −2

Substituting in equation (i), we get

2(1)x² + [2(−2) + 1(3)]xy + 3(−2)y² = 0

i.e., 2x² − xy − 6y² = 0,

Which is the required combined equation.