Question

# If the $$A.M$$. of two numbers a and $$b , a > b > 0$$ is twice their $$G.M..$$ then $$\frac { a } { b } =$$

A
2+3
B
7+23
C
4+23
D
7+43

Solution

## The correct option is B $$7 + 4 \sqrt { 3 }$$Let the larger number be $$x$$ and smaller be $$y$$Given, $$A.M.=2G.M$$$$\Rightarrow\,\dfrac{a+b}{2}=2\sqrt{ab}$$Squaring both sides,we get$$\Rightarrow\,{\left(\dfrac{a+b}{2}\right)}^{2}={\left(2\sqrt{ab}\right)}^{2}$$$$\Rightarrow\,\dfrac{{a}^{2}+2ab+{b}^{2}}{4}=4ab$$$$\Rightarrow\,{a}^{2}+2ab+{b}^{2}=16ab$$$$\Rightarrow\,{a}^{2}+{b}^{2}=16ab-2ab=14ab$$$$\Rightarrow\,{a}^{2}+{b}^{2}-14ab=0$$Dividing by $${b}^{2}$$ throughout we get$$\Rightarrow\,\dfrac{{a}^{2}}{{b}^{2}}+1-14\dfrac{a}{b}=0$$$$\Rightarrow\,\dfrac{{a}^{2}}{{b}^{2}}-14\dfrac{a}{b}+1=0$$$$\Rightarrow\,\dfrac{a}{b}=\dfrac{14\pm\sqrt{196-4\times 1\times 1}}{2}$$$$=\dfrac{14\pm\sqrt{192}}{2}=\dfrac{14\pm\,8\sqrt{3}}{2}=7+4\sqrt{3}$$Mathematics

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