Question

If the slope of one of the straight lines ax² + 2hxy  by² = 0 is thrice that of the other, then show that 3h² = 4ab.

Answer 1

Let m₁ and m₂ be the slope of the pair of straight lines ax² + 2hxy  by² = 0

Then m₁ + m₂ = `(-2"h")/"b"` and m₁m₂ = `"a"/"b"`

Given m₂ = 3m₁ 

∴ m₁ + 3m₁ = `(-2"h")/"b"`

⇒ 4m₁ = `(-2"h")/"b"`

⇒ m₁ = `(-"h")/"2b"`   ....(1)

Also m₁ (3m₁) = `"a"/"b"`

⇒ `3m_1^2 = "a"/"b"`

⇒ `3((-"h")/"2b") = "a"/"b"`   ...[Using (1)]

⇒ `("3h"^2)/("4b"^2) = "a"/"b"`

⇒ `("3h"^2)/("4b") = "a"/1`

⇒ 3h² = 4ab

Hence proved.