Two circles have the following equations $(x +2 )^2+(y -5 )^2= 16$ and $(x +4 )^2+(y +1 )^2= 25$ . 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 · 2018-01-19T13:01:18

Distance between the farthest points of the two circles is

$D_F = R_A + D + R_B = 4 + 6.3246 + 5 = color (brown)(15.3246)$

enter image source here

Standard form of equation of a circle is
$(x - h)^2 + (y-k)^2 = r^2$
Where center coordinates (h,k), r the radius.

Circle 1

$center c_1 (-2,5), radius r_1 = 4$

Circle 2

$center c_2 (-4,-1), radius r_2 = 5$

Distance between the centers

$D = sqrt((-4 - (-2))^2 + (-1-5)^) = sqrt(2^2 + 6^2) = 6.3246$

Since distance between centers greater than their radii, one circle does not contain the other.

Distance between the farthest points of the two circles is

$D_f = r_1 + D + r_2 = 4 + 6.3246 + 5 = color (brown)(15.3246)$
enter image source here

Two circles have the following equations (x +2 )^2+(y -5 )^2= 16 (x +2 )^2+(y -5 )^2= 16 and (x +4 )^2+(y +1 )^2= 25 (x +4 )^2+(y +1 )^2= 25 . 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?