Question

Prove that the product of lengths of perpendiculars drawn from P (x₁, y₁ ) to the lines represented by ax² + 2hxy + by² = 0 is

\(\left|\cfrac{ax_1^2 +2hx_1y_1 +by_1^2}{\sqrt{(a-b)^2 +4h^2}}\right|\)

|(ax₁² + 2hx₁y₁ + by₁²)/√((a-b)² +4h²)|

Answer 1

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

∴ m₁ + m₂ = -\(\cfrac{2h}{b}\) and m₁ m₂ = \(\cfrac{a}{b}\).....(1)

The separate equations of the lines represented by

ax² + 2hxy + by² = 0 are

y = m₁x and y = m₂x

i.e. m₁x – y = 0 and m₂x – y = 0

Length of perpendicular from P(x₁, y₁) on