Question

If (x+y):√xy =4:1 ,find x:y

Answer 1

(x+y)/√(xy) = 4/1
x+y = 4√(xy)  -----------------(1)
(x-y)² = (x+y)² - 4xy
(x-y)² = 16xy - 4xy
(x-y)² = 12xy
x-y = 2√(3xy)  ---------------(2)
(1) + (2)
2x = 2√(xy) (2+√3)
√(x/y) = (2+√3)
x/y = 4+3+4√3
x/y = 7+4√3
x:y = (7+4√3):1

Answer 2

[tex]\frac{x+y}{\sqrt{xy}} = \frac{4}{1} \\ \\ x + y = 4 \sqrt{xy} \ \ \ \ \ - equation\ 1\\ \\ (x + y)^{2} = 4^{2} xy \\ \\ x^{2} + y^{2} + 2 x y = 16 x y \\ \\ x^{2} + y^{2} - 14 x y = 0 \\ \\ (x - y)^{2} = 12 x y \\ \\ x - y = 2 \sqrt{3} \sqrt{ x y} \ \ \ \ \ \ - equation\ 2\\ \\ Solve\ equations\ 1\ and\ 2 \\ \\ [/tex]

[tex]2x = (4+2\sqrt{3})\sqrt{xy} \\ \\ \sqrt{x} = (2+\sqrt{3})\sqrt{y} \\ \\ \frac{\sqrt{x}}{\sqrt{y}} = 2 + \sqrt{3} \\ \\ squaring\ both\ sides: \\ \\ \frac{x}{y} = 4+3+4\sqrt{3} \\ \\ = 7+4\sqrt{3} \\ \\ [/tex]