In a triangle \( A B C, B=30^{\circ}, C=150^{\circ}, b=50 \sqrt{3} . a_{1} \), \( a_{2} \) are t....

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In a triangle \( A B C, B=30^{\circ}, C=150^{\circ}, b=50 \sqrt{3} . a_{1} \),
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\( a_{2} \) are the two values of the third side \( a . x \) and \( y \) are respectively the arithmetic and geometric means of \( a_{1} \) and \( a_{2} \) then \( \left(\frac{x}{\sqrt{3}}\right)^{2}-\left(\frac{y}{\sqrt{6}}\right)^{2} \) is equal to \( (k)^{k} \) where \( k= \)


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