Find-y-x-log-x-y-y-x-below-2x-2-xy-3y-2-0-

Question Number 215519 by walterpieuler last updated on 09/Jan/25

Find {(\frac{y}{x})}^{\log_{\frac{x}{y}} \frac{y}{x}} below :

2 x^{2} + xy - 3 y^{2} = 0

Answered by Rasheed.Sindhi last updated on 10/Jan/25

2 x^{2} + xy - 3 y^{2} = 0

2 (\frac{x}{y}) + 1 - 3 (\frac{y}{x}) = 0

l e t \frac{x}{y} = a

2 a + 1 - \frac{3}{a} = 0

2 a^{2} + a - 3 = 0

(a - 1) (2 a + 3) = 0

a = 1, - \frac{3}{2}

\frac{x}{y} = 1, - \frac{3}{2}

{(\frac{y}{x})}^{\log_{\frac{x}{y}} \frac{y}{x}} = {(\frac{y}{x})}^{- 1} = \frac{x}{y} = 1, - \frac{3}{2}

[∵ \log_{\frac{x}{y}} \frac{y}{x} = \log_{\frac{x}{y}} {(\frac{x}{y})}^{- 1} = - 1]