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If f(x+2y,x-2y)=xy, then f(x,y) equals (A)(x²-y²)÷8 (B)(x²-y²)÷4 (C) (x²+y²)÷4 (D)(x²-y²)/2

If f(x+2y,x-2y)=xy, then f(x,y) equals
(A)(x²-y²)÷8   (B)(x²-y²)÷4   (C) (x²+y²)÷4
(D)(x²-y²)/2

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1 Answers

Arun
25758 Points
3 years ago

I first worked it backwards. I start with f(x,y) = (x^2 - y^2)/8. Then I substitute x+2y for x and x-2y for y. Do the algebra and get xy. This proves the converse of the problem.

Now run it forwards: f(x+2y, x-2y) = xy = 8xy/8 = ( (x^2 + 4xy + 4y^2) - (x^2 - 4xy + 4y^2) ) /8 = ( (x+2y)^2 - (x-2y)^2) )/8. Substitute x for x+2y and y for x-2y and get f(x,y) = ( x^2 - y^2)/8.

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