Cho $\Large a, b\in\mathbb{R}$ thỏa mãn $\Large \log_{4}a=\log_{9}b=\l

Cho $\Large a, b\in\mathbb{R}$ thỏa mãn $\Large \log_{4}a=\log_{9}b=\log_{6} (a-b)$. Tính $\Large M=\dfrac{a}{a+b}$

 

• A.

  A. $\Large M=\dfrac{5+\sqrt{5}}{10}$

   
• B.

  B. $\Large M=\dfrac{\sqrt{5}-1}{2}$

   
• C.

  C. $\Large M=\dfrac{2+\sqrt{3}}{5}$

   
• D.

  D. $\Large M=\dfrac{1}{1+\sqrt{2}}$

Chọn A

Ta có: 

$\Large \log_{4}a=\log_{9}b=\log_{6} (a-b)=t\Rightarrow a=4^t; b=9^t; a-b=6^t\Rightarrow 4^t-9^t=6^t$

$\Large \Leftrightarrow \left(\dfrac{2}{3}\right)^{2t}-\left(\dfrac{2}{3}\right)^t-1=0$ $\Large \Leftrightarrow \left[\begin{align}&\left(\dfrac{2}{3}\right)^t=\dfrac{1-\sqrt{5}}{2}<0, (\text{loại})\\&\left(\dfrac{2}{3}\right)^t=\dfrac{1+\sqrt{5}}{2}\\\end{align}\right.$

$\Large \dfrac{a}{b}=\dfrac{3+\sqrt{5}}{2} \Rightarrow a=b\dfrac{3+\sqrt{5}}{2}$

$\Large \Leftrightarrow M=\dfrac{5+\sqrt{5}}{10}$