If y(3x - y) : x(4x + y) = 5 : 12, find (x² + y²) : (x + y)².

Given, y(3x - y) : x(4x + y) = 5 : 12

$\therefore \dfrac{3xy - y^2}{4x^2 + xy} = \dfrac{5}{12} \\[0.5em] \Rightarrow 12(3xy - y^2) = 5(4x^2 + xy) \\[0.5em] \Rightarrow 36xy - 12y^2 = 20x^2 + 5xy \\[0.5em] \Rightarrow 20x^2 + 12y^2 + 5xy - 36xy = 0 \\[0.5em] \Rightarrow 20x^2 - 31xy + 12y^2 = 0 \\[0.5em]$

Dividing the equation by y²,

$\Rightarrow 20\big(\dfrac{x}{y}\big)^2 - 31\big(\dfrac{x}{y}\big) + 12 = 0 \\[1em] \Rightarrow 20\big(\dfrac{x}{y}\big)^2 - 16\big(\dfrac{x}{y}\big) - 15\big(\dfrac{x}{y}\big) + 12 = 0 \\[1em] \Rightarrow 4\big(\dfrac{x}{y}\big)(5\big(\dfrac{x}{y}\big) - 4) - 3(5\big(\dfrac{x}{y}\big) - 4) = 0 \\[1em] \Rightarrow (5\big(\dfrac{x}{y}\big) - 4)(4\big(\dfrac{x}{y}\big) - 3) = 0 \\[1em] \Rightarrow 5\big(\dfrac{x}{y}\big) - 4 = 0 \text{ or } 4\big(\dfrac{x}{y}\big) - 3 = 0 \\[1em] \Rightarrow \dfrac{x}{y} = \dfrac{4}{5} \text{ or } \dfrac{x}{y} = \dfrac{3}{4}.$

We have to find value of (x² + y²) : (x + y)².

$= (x^2 + y^2) : (x^2 + y^2 + 2xy) \\[0.5em] = \dfrac{x^2 + y^2}{x^2 + y^2 + 2xy} \\[0.5em]$

Dividing numerator and denominator by y²,

$= \dfrac{\dfrac{x^2 + y^2}{y^2}}{\dfrac{x^2 + y^2 + 2xy}{y^2}} \\[1em] = \dfrac{\big(\dfrac{x}{y}\big)^2 + 1}{\big(\dfrac{x}{y}\big)^2 + 1 + 2\big(\dfrac{x}{y}\big)} \\[1em]$

Putting value of $\dfrac{x}{y} = \dfrac{4}{5}$,

$\dfrac{\big(\dfrac{4}{5}\big)^2 + 1}{\big(\dfrac{4}{5}\big)^2 + 1 + 2\big(\dfrac{4}{5}\big)} \\[1em] = \dfrac{\big(\dfrac{16}{25}\big) + 1}{\big(\dfrac{16}{25}\big) + 1 + \big(\dfrac{8}{5}\big)} \\[1em] = \dfrac{\dfrac{16 + 25}{25}}{\dfrac{16 + 25 + 40}{25}} \\[1em] = \dfrac{41}{81} \\[1em] = 41 : 81 \\[1em]$

Putting value of $\dfrac{x}{y} = \dfrac{3}{4}$,

$\dfrac{\big(\dfrac{3}{4}\big)^2 + 1}{\big(\dfrac{3}{4}\big)^2 + 1 + 2\big(\dfrac{3}{4}\big)} \\[1em] = \dfrac{\big(\dfrac{9}{16}\big) + 1}{\big(\dfrac{9}{16}\big) + 1 + \big(\dfrac{6}{4}\big)} \\[1em] = \dfrac{\dfrac{9 + 16}{16}}{\dfrac{9 + 16 + 24}{16}} \\[1em] = \dfrac{25}{49} \\[1em] = 25 : 49$

Hence, the value of ratio (x² + y²) : (x + y)² is 41 : 81 or 25 : 49.