Two circles have the following equations: ${(x - 1)}^{2} + {(y - 4)}^{2} = 64$ $(x -1 )^2+(y -4 )^2= 64$ and ${(x + 6)}^{2} + {(y - 9)}^{2} = 49$ $(x +6 )^2+(y -9 )^2= 49$ . 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?

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Answer 1 · 2017-03-16T09:20:44

$As r_{r} + r_{b} is greater than l, one circle contains the other.$ $"As "r_r+r_b " is greater than l, one circle contains the other."$

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$The general Circle equation can be written as the below$ $"The general Circle equation can be written as the below"$

${(x - a)}^{2} + {(y - b)}^{2} = r^{2}$ $(x-a)^2+(y-b)^2=r^2$

$Where (a,b) center coordinates,r radius of circle.$ $"Where (a,b) center coordinates,r radius of circle."$

$O_{b} (- 6, 9) represents the center of blue circle.$ $O_b(-6,9) " represents the center of blue circle."$

$O_{r} (1, 4) represents the center of red circle.$ $O_r(1,4)" represents the center of red circle."$

$r_{r} : radius of the red circle , r_{r} = \sqrt{64}, r_{r} = 8 units$ $r_r:" radius of the red circle ,"r_r=sqrt(64)" , "r_r=8 " units"$

$r_{b} : radius of the red blue , r_{r} = \sqrt{49}, r_{b} = 7 units$ $r_b:" radius of the red blue ,"r_r=sqrt(49)" , "r_b=7 " units"$

$l : represents distance from O_{r} to O_{b}$ $l : "represents distance from "O_r " to "O_b$

$l = \sqrt{{(1 + 6)}^{2} + {(9 - 4)}^{2}}), l = \sqrt{7^{2} + 5^{2}}, l = 8.6$ $l=sqrt((1+6)^2+(9-4)^2))" , "l=sqrt(7^2+5^2)" , "l=8.6"$

$r_{r} + r_{b} = 7 + 8 = 15$ $r_r+r_b=7+8=15$

$l = 8.6$ $l=8.6$

$r_{r} + r_{b} > l$ $r_r+r_b >l$

$As r_{r} + r_{b} is greater than l, one circle contains the other.$ $"As "r_r+r_b " is greater than l, one circle contains the other."$

Two circles have the following equations: (x−1)2+(y−4)2=64(x -1 )^2+(y -4 )^2= 64 and (x+6)2+(y−9)2=49(x +6 )^2+(y -9 )^2= 49 . 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?