If 10x² − 23xy + 9y² = 0, find x : y.

Given,

10x² − 23xy + 9y² = 0

Dividing both sides by xy we get,

$\Rightarrow \dfrac{10x^2 − 23xy + 9y^2}{xy} = 0 \\[1em] \Rightarrow 10\dfrac{x}{y} - 23 + 9\dfrac{y}{x} = 0 \\[1em] \Rightarrow 10\dfrac{x}{y} + 9\dfrac{y}{x} = 23$

Let, $\dfrac{x}{y}$ = t

$\Rightarrow 10t + 9\dfrac{1}{t} = 23 \\[1em] \Rightarrow \dfrac{10t^2 + 9}{t} = 23 \\[1em] \Rightarrow 10t^2 + 9 = 23t \\[1em] \Rightarrow 10t^2 - 23t + 9 = 0 \\[1em] \Rightarrow 10t^2 - 5t - 18t + 9 = 0 \\[1em] \Rightarrow 5t(2t - 1) - 9(2t - 1) = 0 \\[1em] \Rightarrow (5t - 9)(2t - 1) = 0 \\[1em] \Rightarrow 5t - 9 = 0 \text{ or } 2t - 1 = 0 \\[1em] \Rightarrow 5t = 9 \text{ or } 2t = 1 \\[1em] \Rightarrow t = \dfrac{9}{5} \text{ or } t = \dfrac{1}{2} \\[1em] \Rightarrow \dfrac{x}{y} = \dfrac{9}{5} \text{ or } \dfrac{x}{y} = \dfrac{1}{2}$

Hence, x : y = 9 : 5 or 1 : 2.