If $(x + y) : (x - y) = 4 : 1$ , then find the ratio $(x^{2} + y^{2}) : (x^{2} - y^{2})$ .

13 : 8

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15 : 8

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17 : 8

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19 : 8

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Solution

The correct option is C $17 : 8$
Given $\frac{x + y}{x - y} = \frac{4}{1} \Rightarrow x + y = 4 x - 4 y \Rightarrow 3 x = 5 y \Rightarrow \frac{x}{y} = \frac{5}{3}$
$∴ \frac{x^{2} + y^{2}}{x^{2} - y^{2}} = \frac{\frac{x^{2}}{y^{2}} + 1}{\frac{x^{2}}{y^{2}} - 1} = \frac{\frac{25}{9} + 1}{\frac{25}{9} - 1} = \frac{\frac{34}{9}}{\frac{16}{9}} = \frac{34}{16} = \frac{17}{8}$
$\Rightarrow (x^{2} + y^{2}) : (x^{2} + y^{2}) = 17 : 8$

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