If x⁸ + 25 = \(\frac{x^4}{y^2}\) (24y² - y⁴ - 49)

Find the value of \(\frac{2x^4 - y^2}{x^4y^2}\).

\(\frac{3}{35}\)

Divide the given equation by x⁴

\(\frac{x^8}{x^4}\) + \(\frac{25}{x^4}\) = \(\frac{x^4}{y^2 × x^4}\) (24y² - y⁴ - 49)

⇒ x⁴ + \(\frac{25}{x^4}\) = 1 (24 - y² - \(\frac{49}{y^2}\))

⇒ x⁴ + \(\frac{25}{x^4}\) + y² + \(\frac{49}{y^2}\)) - 24 = 0

⇒ (x² - \(\frac{5}{x^2}\))² + (y - \(\frac{7}{y}\))² = 0

x² - \(\frac{5}{x^2}\) = 0, x⁴ = 5

y - \(\frac{7}{y}\) = 0

^y2 = 7

Put in \(\frac{2x^4 - y^2}{x^4y^2}\)

⇒ \(\frac{2 × 5 - 7}{5 × 7}\) = \(\frac{3}{35}\)