# Two circles have the following equations #(x -4 )^2+(y +3 )^2= 9 # and #(x +4 )^2+(y -1 )^2= 16 #. Does one circle contain the other? If not, what is the greatest possible distance between a point on one circle and another point on the other?

Circles are outside each other and greatest possible distance between a point on one circle and another point on the other is

graph{(x^2+y^2-8x+6y+16)(x^2+y^2+8x-2y+1)=0 [-10.5, 9.5, -5.665, 4.755]}

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The circles described by the given equations do not contain each other. The greatest possible distance between a point on one circle and another point on the other can be found by calculating the sum of their radii. In this case, the radii of the circles are (3) and (4) respectively. Therefore, the greatest possible distance between a point on one circle and another point on the other is (3 + 4 = 7).

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