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Question
Find the fourth proportional to:
x3 - y2, x4 + x2y2 + y4, x - y.
Solution
Let A be the fourth proportional then
x3 - y2 : x4 + x2y2 + y4 = x - y : A
⇒ `(x^3 - y^3)/(x^4 + x^2y^2 + y^4) = (x - y)/"A"`
⇒ A(x3 - y3) = (x - y)(x4 + x2y2 + y4)
⇒ A = `((x - y)(x^4 + x^2y^2 + y^4))/(x^3 - y^3)`
⇒ A = `((x - y)(x^2 + y^2 + xy)(x^2 + y^2 - xy))/((x - y)(x^2 + xy + y^2)`
⇒ A = x2 + y2 - xy.
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