Solve {l}{text{If}logx+y/7=1/2(logx+logy);text{thenshowthat}x^2/y+2/x=47}{text{If}logfrac{a+b}{3}=frac{1}{2}(loga+logb);text{thenprovethat}a^2+b^2=7ab}

$$\left. \begin{array} { l } { \text { If } \log \frac { x + y } { 7 } = \frac { 1 } { 2 } ( \log x + \log y ) ; \text { then show that } \frac { x ^ { 2 } } { y } + \frac { 2 } { x } = 47 } \\ { \text { If } \log \frac { a + b } { 3 } = \frac { 1 } { 2 } ( \log a + \log b ) ; \text { then prove that } a ^ { 2 } + b ^ { 2 } = 7 a b } \end{array} \right.$$

Solve for x, y, a, b

$$x=\frac{\left(987\sqrt{5}+2207\right)\left(-\sqrt{2\left(4045807-1809339\sqrt{5}\right)}+2209-987\sqrt{5}\right)}{8}\approx 0.042554118\text{, }y=\frac{\left(23184\sqrt{5}+51841\right)\left(-\sqrt{2\left(4045807-1809339\sqrt{5}\right)}+2209-987\sqrt{5}\right)}{4}\approx 1.999137745\text{, }a=\frac{\left(7-3\sqrt{5}\right)b}{2}\text{, }b>0$$

$$x=\frac{\left(987\sqrt{5}+2207\right)\left(-\sqrt{2\left(4045807-1809339\sqrt{5}\right)}+2209-987\sqrt{5}\right)}{8}\approx 0.042554118\text{, }y=\frac{\left(23184\sqrt{5}+51841\right)\left(-\sqrt{2\left(4045807-1809339\sqrt{5}\right)}+2209-987\sqrt{5}\right)}{4}\approx 1.999137745\text{, }a=\frac{\left(3\sqrt{5}+7\right)b}{2}\text{, }b>0$$

$$x=\frac{\left(987\sqrt{5}+2207\right)\left(\sqrt{2\left(4045807-1809339\sqrt{5}\right)}+2209-987\sqrt{5}\right)}{8}\approx 2207.956992778\text{, }y=\frac{\left(23184\sqrt{5}+51841\right)\left(\sqrt{2\left(4045807-1809339\sqrt{5}\right)}+2209-987\sqrt{5}\right)}{4}\approx 103726.979566367\text{, }a=\frac{\left(7-3\sqrt{5}\right)b}{2}\text{, }b>0$$

$$x=\frac{\left(987\sqrt{5}+2207\right)\left(\sqrt{2\left(4045807-1809339\sqrt{5}\right)}+2209-987\sqrt{5}\right)}{8}\approx 2207.956992778\text{, }y=\frac{\left(23184\sqrt{5}+51841\right)\left(\sqrt{2\left(4045807-1809339\sqrt{5}\right)}+2209-987\sqrt{5}\right)}{4}\approx 103726.979566367\text{, }a=\frac{\left(3\sqrt{5}+7\right)b}{2}\text{, }b>0$$

$$x=\frac{\left(2207-987\sqrt{5}\right)\left(-\sqrt{2\left(1809339\sqrt{5}+4045807\right)}+987\sqrt{5}+2209\right)}{8}\approx 0.044535724\text{, }y=\frac{\left(51841-23184\sqrt{5}\right)\left(-\sqrt{2\left(1809339\sqrt{5}+4045807\right)}+987\sqrt{5}+2209\right)}{4}\approx 0.000947997\text{, }a=\frac{\left(7-3\sqrt{5}\right)b}{2}\text{, }b>0$$

$$x=\frac{\left(2207-987\sqrt{5}\right)\left(-\sqrt{2\left(1809339\sqrt{5}+4045807\right)}+987\sqrt{5}+2209\right)}{8}\approx 0.044535724\text{, }y=\frac{\left(51841-23184\sqrt{5}\right)\left(-\sqrt{2\left(1809339\sqrt{5}+4045807\right)}+987\sqrt{5}+2209\right)}{4}\approx 0.000947997\text{, }a=\frac{\left(3\sqrt{5}+7\right)b}{2}\text{, }b>0$$

$$x=\frac{\left(2207-987\sqrt{5}\right)\left(\sqrt{2\left(1809339\sqrt{5}+4045807\right)}+987\sqrt{5}+2209\right)}{8}\approx 0.95591738\text{, }y=\frac{\left(51841-23184\sqrt{5}\right)\left(\sqrt{2\left(1809339\sqrt{5}+4045807\right)}+987\sqrt{5}+2209\right)}{4}\approx 0.020347874\text{, }a=\frac{\left(7-3\sqrt{5}\right)b}{2}\text{, }b>0$$

$$x=\frac{\left(2207-987\sqrt{5}\right)\left(\sqrt{2\left(1809339\sqrt{5}+4045807\right)}+987\sqrt{5}+2209\right)}{8}\approx 0.95591738\text{, }y=\frac{\left(51841-23184\sqrt{5}\right)\left(\sqrt{2\left(1809339\sqrt{5}+4045807\right)}+987\sqrt{5}+2209\right)}{4}\approx 0.020347874\text{, }a=\frac{\left(3\sqrt{5}+7\right)b}{2}\text{, }b>0$$