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The line x + 3y = 0 is a diameter of the circle x2 + y2 + 6x + 2y = 0.

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False  

Given equation of the circle is x2+y2+6x+2y=0  

 Centre=(-3,-1) 

 Clearly, it does not lie on the line x+3y=0  as-3+3(-1)=-6   

So this line is not the diameter of the circle

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