Question

If (x + y)³ − (x − y)³ − 6y(x² − y²) = ky², then k =

Options

• 1

• 2

• 4

• 8

Answer 1

The given equation is

(x + y)³ − (x − y)³ − 6y(x² − y²) = ky²

Recall the formula 

`a^3 - b^3 = (a-b)(a^2 +ab +b^2)`

Using the above formula, we have

`(x+y)^3 - (x-y)^3  - 6y(x^2 -y^2 )ky^2`

`⇒ {(x+y)^3 - (x-y)^3} - 6y (x^2 - y^2) = ky^2`

` ⇒ 2y(x^2 + 2xy + y^2 +x^2 - y^2 - x^2 - 2xy +y^2) -6y(x^2 - y^2) = ky^3`

`⇒ 2y(3x^2 +y^2) -6y(x^2 - y^2) = ky^3`

`⇒6x^2y +2y^3 - 6x^2 y +6y^3 = ky^3`

`⇒ 8y^3 = ky^3`

 `⇒ ky^3 = 8y^3`

⇒ k =8, provided y ≠0.