Q4
5 marks
Generalised Binomial Theorem
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On January 3, 2019, the Chang'e-4 probe successfully achieved humanity's first soft landing on the far side of the moon. A key technical challenge in achieving soft landing on the far side of the moon is maintaining communication between the ground and the probe. To solve this problem, the Chang'e-4 relay satellite ``Queqiao'' was launched, which orbits around the Earth-Moon Lagrange point $L _ { 2 }$. The $L _ { 2 }$ point is an equilibrium point located on the extension of the Earth-Moon line. Let the mass of Earth be $M _ { 1 }$, the mass of the Moon be $M _ { 2 }$, the Earth-Moon distance be $R$, and the distance from the $L _ { 2 }$ point to the Moon be $r$. According to Newton's laws of motion and the law of universal gravitation, $r$ satisfies the equation: $\frac { M _ { 1 } } { ( R + r ) ^ { 2 } } + \frac { M _ { 2 } } { r ^ { 2 } } = ( R + r ) \frac { M _ { 1 } } { R ^ { 3 } }$.
Let $\alpha = \frac { r } { R }$. Since $\alpha$ is very small, in approximate calculations $\frac { 3 \alpha ^ { 3 } + 3 \alpha ^ { 4 } + \alpha ^ { 5 } } { ( 1 + \alpha ) ^ { 2 } } \approx 3 \alpha ^ { 3 }$. Then the approximate value of $r$ is
A．$\sqrt { \frac { M _ { 2 } } { M _ { 1 } } } R$
B．$\sqrt { \frac { M _ { 2 } } { 2 M _ { 1 } } R }$
C．$\sqrt [ 3 ] { \frac { 3 M _ { 2 } } { M _ { 1 } } R }$
D．$\sqrt [ 3 ] { \frac { M _ { 2 } } { 3 M _ { 1 } } } R$