1.

If logx (a – b) – logx (a + b) = logx (\(\frac{b}{a}\)), find \(\frac{a^2}{b^2}+\frac{b^2}{a^2}\) .

Answer»

Given, logx (a – b) – logx (a + b) = logx (\(\frac{b}{a}\)) ⇒ logx \(\bigg[\frac{(a-b)}{(a+b)}\bigg]\) = logx (\(\frac{b}{a}\))

⇒ a(a – b) = b(a + b) ⇒ a2 – ab = ab + b2

⇒ a2 – b2 = 2ab ⇒ a2 – 2ab – b2 = 0 ⇒ \(\big(\frac{a}{b}\big)^2\) - 2\(\big(\frac{a}{b}\big)\) -1 = 0

This is a quadratic equation in \(\frac{a}{b}\) and the product of the roots is –1 i.e, if \(\frac{a}{b}\) is a root, then \(\big(-\frac{a}{b}\big)\) b a is the other root. Also, sum of its roots = 2

∴ \(\big(\frac{a}{b}\big)^2\) + \(\big(\frac{a}{b}\big)^2\) = \(\frac{a^2}{b^2}\) + \(\frac{b^2}{a^2}\) = \(\bigg[\frac{a}{b}+\big(-\frac{b}{a}\big)\bigg]^2\) + 2 = 22 + 2 = 6.


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