Verified answer

[tex] \frac{7 + \sqrt{5} }{7 - \sqrt{5} } - \frac{7 - \sqrt{5} }{7 + \sqrt{5} } = a + \frac{7}{11} \sqrt{5} b \\ \\ \frac{ {(7 + \sqrt{5} })^{2} - ( {7 - \sqrt{5} })^{2} }{49 - 5} = a + \frac{7}{11} \sqrt{5} b \\ \\ \frac{4.7. \sqrt{5} }{44} = a + \frac{7}{11} \sqrt{5} b \\ \\ \frac{7}{11} \sqrt{5} = a + \frac{7}{11} \sqrt{5} b[/tex]

Comparing both sides we get

[tex]a = 0 \: \: \: \: and \: \: \: b = 1[/tex]