If $a^{2} + b^{2} = 23 a b (a, b > 0)$ , then A.M. of Ina and lnb is equal to

\frac{1}{2} I n (a + b)

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(\frac{a + b}{25})

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(\frac{a + b}{5})

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I n (\frac{a - b}{5})

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Solution

The correct option is D $I n (\frac{a - b}{5})$

$a^{2} + b^{2} = 23 a b \Rightarrow (a - b)^{2} = 25 a b \Rightarrow a - b = 5 \sqrt{a b} \Rightarrow \frac{a - b}{5} = \sqrt{a b}$
SO, $l n (\sqrt{a b}) = l n (\frac{a - b}{5}) \Rightarrow l n (a b) = 2 l n (\frac{a - b}{5}) \Rightarrow \frac{l n a + l n b}{2} = l n (\frac{a - b}{5})$

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If a2+b2=23ab(a,b>0), then A.M. of Ina and lnb is equal to

The correct option is D In(a−b5)a2+b2=23ab⇒(a−b)2=25ab⇒a−b=5√ab⇒a−b5=√abSO, ln(√ab)=ln(a−b5)⇒ln(ab)=2ln(a−b5)⇒lna+lnb2=ln(a−b5)

If $a^{2} + b^{2} = 23 a b (a, b > 0)$ , then A.M. of Ina and lnb is equal to