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Question
Give `(x^3 + 12x)/(6x^2 + 8) = (y^3 + 27y)/(9y^2 + 27)` Using componendo and dividendo find x : y.
Solution
Give `(x^3 + 12x)/(6x^2 + 8) = (y^3 + 27y)/(9y^2 + 27)`
Using componendo-dividendo, we have
`(x^3 + 12x + 6x^2 + 8)/(x^3 + 12x - 6x^2 - 8) = (y^3 + 27y + 9y^2 + 27)/(y^3 + 27y 9y^2 - 27)`
⇒ `(x + 2)^3/(x 2)^3 = (y + 3)^3/(9y - 3)^3`
⇒ `((x + 2)/(x - 2))^3 = ((y + 3)/(y - 3))^3`
⇒ `(x + 2)/(x - 2) = (y + 3)/(y - 3)`
Again using componendo-dividendo, we get
`(x + 2 + x - 2)/(x + 2 - x + 2) = (y + 3 + y - 3)/(y + 3 - y + 3)`
⇒ `(2x)/(4) = (2y)/(3)`
⇒ `x/(2) = y/(3)`
⇒ `x/y = (2)/(3)`
Thus the required ratio is x : y = 2 : 3.
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