If (4x² − 3y²) : (2x² + 5y²) = 12 : 19 then x : y is:

1. 2 : 3

2. 1 : 2

3. 2 : 1

4. 3 : 2

Given,

$\Rightarrow \dfrac{4x^2 - 3y^2}{2x^2 + 5y^2} = \dfrac{12}{19} \\[1em] \Rightarrow 19(4x^2 - 3y^2) = 12(2x^2 + 5y^2) \\[1em] \Rightarrow 76x^2 - 57y^2 = 24x^2 + 60y^2 \\[1em] \Rightarrow 76x^2 - 24x^2 = 60y^2 + 57y^2 \\[1em] \Rightarrow 52x^2 = 117y^2 \\[1em] \Rightarrow \dfrac{x^2}{y^2} = \dfrac{117}{52} \\[1em] \Rightarrow \dfrac{x^2}{y^2} = \dfrac{9}{4} \\[1em] \Rightarrow \dfrac{x}{y} = \sqrt{\dfrac{9}{4}} \\[1em] \Rightarrow \dfrac{x}{y} = \dfrac{3}{2}.$

Thus,

x : y = 3 : 2.

Hence, option 4 is the correct option.