If a and b be coprime numbers and then find the minimum value of (a2 + b2).

Engineering

Mathematics

Exponential and Logarithmic Limits

Question

If a and b be coprime numbers and $\underset{x \to \frac{π^{-}}{2}}{Lim} (1 + tanx) ((1 + tanx) \log (\frac{1 + tanx}{2 + tanx}) + 1) = \frac{a}{b}$ then find the minimum value of (a² + b²).

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$x \to \frac{π^{-}}{2}, t = tanx \to \infty$

$= \underset{t \to \infty}{Lim} (1 + t) {(1 + t) \log (1 - \frac{1}{2 + t}) + 1}$

$= \underset{t \to \infty}{Lim} (1 + t) [(1 + t) {- \frac{1}{(2 + t)} - \frac{1}{2 {(2 + t)}^{2}} - \frac{1}{3 {(2 + t)}^{3}} - . ......} + 1]$

$= \underset{t \to \infty}{Lim} (t + 1) [{- \frac{1 + t}{t + 2} - \frac{(1 + t)}{2 {(t + 2)}^{2}} - \frac{(1 + t)}{3 {(t + 2)}^{3}} - . ......} + 1]$

$= \underset{t \to \infty}{Lim} (t + 1) {\frac{1}{t + 2} - \frac{(1 + t)}{2 {(t + 2)}^{2}} - \frac{(t + 1)}{3 {(t + 2)}^{3}} . .........}$

$= \underset{t \to \infty}{Lim} {\frac{t + 1}{t + 2} - \frac{{(t + 1)}^{2}}{2 {(t + 2)}^{2}} - \frac{{(t + 1)}^{2}}{3 {(t + 2)}^{3}} . ......}$

$= 1 - \frac{1}{2} = \frac{1}{2} \equiv \frac{a}{b}$

a = k, b = 2k

a² + b² = 5k²

a² + b² |_min. = 5