Question

If x and y both are positive and (2x² – 5y²) : xy = 1 : 3, find x : y.

Answer 1

(2x² – 5y²) : xy = 1 : 3

`\implies (2x^2 - 5y^2)/(xy) = 1/3`

`\implies (2x)/y - (5y)/x = 1/3`

Put `x/y = a` we get

`\implies 2a - 5 1/a = 1/3`

`\implies` 3(2a² – 5) = a

`\implies` 6a² – a – 15 = 0

`\implies` 6a² + 9a – 10a – 15 = 0

`\implies` 3a(2a + 3) – 5(2a + 3) = 0

`\implies` (2a + 3)(3a – 5) = 0

`\implies` (2a + 3) = 0 or (3a – 5) = 0

`\implies a = -3/2` or `a = 5/3`

`a = -3/2` is not acceptable, as x and y both are positive.

`a = 5/3`

`\implies x/y = 5/3`

`\implies` x : y = 5 : 3