Question

Find the fourth proportion to the following : 

(x² - y²),(x³ + y³)anc(x³ - xy² + x²y- y³) 

Answer 1

Let z be the fourth proportion 

(x² -y³) (x³ + y³) :: (x³ - xy² + x²y - y³) : z 

⇒ (x² - y² ) z = (x³ + y³)(x³- xy² + x²y -y³) 

⇒ z = `(("x"^3 + "y"^3)("x"^3 - "xy"^2 + "x"^2"y" - "y"^3))/("x"^2 - "y"^2)`

⇒ z = `(("x" + "y")("x"^2 - "xy"^2 + "x"^2"y" - "y"^3))/(("x + y")("x - y"))`

⇒ z = (x² - xy + y²)(x + y)²

Therefore . the fourth proportion is (x² + y² - xy)(x + y)²