Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 543

\[\boxed{\mathbf{543}.}\]

\[\frac{\left( x^{2} + 9 \right)^{- 0,5} + \left( x^{2} - 9 \right)^{- 0,5}}{\left( x^{2} + 9 \right)^{- 0,5} - \left( x^{2} - 9 \right)^{- 0,5}} =\]

\[= \left( \frac{1}{\sqrt{x^{2} + 9}} + \frac{1}{\sqrt{x^{2} - 9}} \right)\ :\]

\[:\left( \frac{1}{\sqrt{x^{2} + 9}} - \frac{1}{\sqrt{x^{2} - 9}} \right) =\]

\[= \frac{\sqrt{x^{2} - 9} + \sqrt{x^{2} + 9}}{\sqrt{\left( x^{2} + 9 \right)\left( x^{2} - 9 \right)}}\ :\]

\[:\frac{\sqrt{x^{2} - 9} - \sqrt{x^{2} + 9}}{\sqrt{\left( x^{2} + 9 \right)\left( x^{2} - 9 \right)}} =\]

\[= \frac{\sqrt{x^{2} - 9} + \sqrt{x^{2} + 9}}{\sqrt{\left( x^{2} + 9 \right)\left( x^{2} - 9 \right)}} \cdot\]

\[\cdot \frac{\sqrt{\left( x^{2} + 9 \right)\left( x^{2} - 9 \right)}}{\sqrt{x^{2} - 9} - \sqrt{x^{2} + 9}} =\]

\[= \frac{\sqrt{x^{2} - 9} + \sqrt{x^{2} + 9}}{\sqrt{x^{2} - 9} - \sqrt{x^{2} + 9}}\]

\[x = 3 \cdot \left( \frac{a^{2} + b^{2}}{2ab} \right)^{\frac{1}{2}}:\]

\[1)\ a > 0;\ \ b > 0;\ \ a > b:\]

\[\frac{a - b + a + b}{a - b - a - b} = \frac{2a}{- 2b} = - \frac{a}{b}.\]

\[2)\ a > 0;\ \ b > 0;\ \ b > a:\]

\[\frac{- a + b + a + b}{- a + b - a - b} = \frac{2b}{- 2a} = - \frac{b}{a}.\]

\[3)\ a < 0;\ \ b < 0;b > a:\]

\[\frac{a - b + a + b}{a - b - a - b} = \frac{2a}{- 2b} = - \frac{a}{b}.\]

\[4)\ a < 0;b < 0;a > b:\]

\[\frac{- a + b + a + b}{- a + b - a - b} = \frac{2b}{- 2a} = - \frac{b}{a}.\]