Question

If the slope of one of the lines given by ax² + 2hxy + by² = 0 is three times the other, prove that 3h² = 4ab.

Answer 1

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

We are given that m₂ = 3m₁ 

∴ ${\displaystyle {\text{m}}_1+3{\text{m}}_1=-\frac{\text{2h}}{\text{b}}}$

∴ ${\displaystyle 4{\text{m}}_1=-\frac{\text{2h}}{\text{b}}}$

∴ ${\displaystyle {\text{m}}_1=-\frac{\text{h}}{\text{2b}}}$

Also, m₁(3m₁) = ${\displaystyle \frac{\text{a}}{\text{b}}}$

∴ ${\displaystyle 3{\text{m}}_1^2=\frac{\text{a}}{\text{b}}}$

∴ ${\displaystyle 3{(-\frac{\text{h}}{\text{2b}})}^2=\frac{\text{a}}{\text{b}}}$      ....[By (1)]

∴ ${\displaystyle \frac{{\text{3h}}^2}{{\text{4b}}^2}=\frac{\text{a}}{\text{b}}}$

∴ 3h² = 4ab, as b ≠ 0