Question

Prove that the acute angle θ between the lines represented by the equation ax² + 2hxy+ by² = 0 is tanθ = `|(2sqrt(h^2 - ab))/(a + b)|` Hence find the condition that the lines are coincident.

Answer 1

Let m₁ and m₂ be slopes of lines represented by the equation

ax² + 2hxy + by² = 0.

∴ `m_1 + m_2 = (-2h)/b and m_1 m_2 = a/b`

∴ `(m_1 - m_2)^2 = (m_1 + m_2)^2 - 4m_1 m_2`

= `((2h)/b)^2 - 4(a/b)`

= `(4h^2)/b^2 - (4a)/b`

= `(4h^2 - 4ab)/b^2`

= `(4(h^2 - ab))/b^2`

∴ `m_1 - m_2 = ± (2sqrt(h^2 - ab))/b`

As θ is the acute angle between the lines, then: 

`tan theta = |(m_1 - m_2)/(1 + m_1m_2)|`

`= |((2sqrt(h^2 - ab))/(b))/(1 + a/b)|`

`tan theta = |(2sqrt(h^2 - ab))/(a + b)|`

Now, if the lines are coincident,

then θ = 0

tan θ = 0

Lines represented by ax² + 2hxy + by² = 0 are coincident if and only if m₁ = m₂

∴ m₁ - m₂ = 0

∴ `(2sqrt(h^2 - ab))/b = 0`

∴ `h^2 - ab = 0`

∴ `h^2 = ab`

Lines represented by ax² + 2hxy + by² = 0 are coincident if and only if h² = ab.