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Solve this question 5x2 3y2) : xy II : 2 , find y : x.

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Please find below the solution to the asked query:

We have : ( 5 x² - 3 y² ) : x y = 11 : 2 , So

$$\begin{gathered} \Rightarrow \frac{\left( 5 x^2 - 3 y^2 \right)}{x y}= \frac{11}{2} \\ \Rightarrow 2\left( 5 x^2 - 3 y^2 \right)= 11 x y \\ \Rightarrow 10 x^2 - 6 y^2= 11 x y \\ \Rightarrow x^2\left(10 -\frac{ 6 y^2}{ x^2}\right)= 11 x y \\ \Rightarrow 10 -\frac{ 6 y^2}{ x^2}= 11 \frac{y}{x} \\ \Rightarrow 10 -6 k^2= 11 k \left[\mathrm{We} \mathrm{assume} k = \frac{y}{x}\right] \\ \Rightarrow 10 -6 k^2 - 11 k = 0 \\ \Rightarrow 6 k^2 + 11 k - 10 = 0 \\ \Rightarrow 6 k^2 + 15 k - 4 k - 10 = 0 \left[\mathrm{Splitting} \mathrm{the} \mathrm{middle} \mathrm{term} \mathrm{method} \right] \\ \Rightarrow 3 k\left( 2 k+ 5 \right)- 2 \left( 2 k+ 5 \right) = 0 \\ \Rightarrow \left(3 k- 2 \right)\left( 2 k+ 5 \right) = 0 \\ \mathrm{So}, \\ k = \frac{2}{3} , \frac{y}{x} = \frac{2}{3} \\ \mathrm{And} \\ k =- \frac{5}{2} , \frac{y}{x} = - \frac{5}{2} \\ \mathrm{we} \mathrm{assume} \mathrm{x} \mathrm{and} \mathrm{y} \mathrm{are} \mathrm{positive} \mathrm{numbers} , \mathrm{So} \mathit{y}\mathbf{ }\mathbf{:}\mathbf{ }\mathit{x}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{2}\mathbf{ }\mathbf{:}\mathbf{ }\mathbf{3} \end{gathered}$$

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