Question

If the slope of one of the lines given by ax² + 2hxy + by² = 0 is four times the other, show that 16h² = 25ab.

Answer 1

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

∴ m₁ + m₂ = ${\displaystyle -\frac{\text{2h}}{\text{b}}}$ 

and m₁m₂ = ${\displaystyle \frac{\text{a}}{\text{b}}}$

We are given that m₂ = 4m₁

∴ m₁ + 4m₁ = ${\displaystyle -\frac{\text{2h}}{\text{b}}}$

∴ 5m₁ = ${\displaystyle \frac{-\text{2h}}{\text{b}}}$

∴ m₁ = ${\displaystyle -\frac{\text{2h}}{\text{5b}}}$    .....(1)

Also, m₁(4m₁) = ${\displaystyle \frac{\text{a}}{\text{b}}}$

∴ ${\displaystyle 4{\text{m}}_1^2=\frac{\text{a}}{\text{b}}}$

∴ ${\displaystyle {\text{m}}_1^2=\frac{\text{a}}{\text{4b}}}$

∴ ${\displaystyle {\left(\frac{\text{- 2h}}{\text{5b}}\right)}^2=\frac{\text{a}}{\text{4b}}}$     ...[By(1)]

∴ ${\displaystyle \frac{{\text{4h}}^2}{{\text{25b}}^2}=\frac{\text{a}}{\text{4b}}}$

∴ ${\displaystyle \frac{{\text{4h}}^2}{\text{25b}}=\frac{\text{a}}{4}}$, as b ≠ 0

∴ 16h² = 25ab

This is the required condition.