If x8 + 25 = \(\frac{x^4}{y^2}\) (24y2 - y4 - 49)
Find the value of \(\frac{2x^4 - y^2}{x^4y^2}\).
Answer & explanation
Correct answer: option 3
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 \(\frac{2x^4 - y^2}{x^4y^2}\)
⇒ \(\frac{2 × 5 - 7}{5 × 7}\) = \(\frac{3}{35}\)