Two circles have the following equations: #(x +6 )^2+(y -5 )^2= 64 # and #(x -9 )^2+(y +4 )^2= 81 #. 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?

Question

Two circles have the following equations: #(x +6  )^2+(y -5   )^2= 64  # and #(x  -9   )^2+(y +4  )^2= 81 #.  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?

Aaliyah Anderson · Accepted Answer

The circles overlap but the small circle isn't contained in the big one. The biggest possible distance between 2 points in the 2 circles is #sqrt226+17~=32.033# #(x+6)^2+(y-5)^2=64# => #r_1=8#, #C_1 (-6,5)# #(x-9)^2+(y-4)^2=81# => #r_2=9#, #C_2 (9,4)# #r_1+r_2=17# #r_2-r_1=1# #d_(C_1C_2)=sqrt((9+6)^2+(4-5)^2)=sqrt(225+1)=sqrt(226)~=15.033# Since #r_2-r_1 < d_(C_1C_2) < r_1+r_2# the circles overlap but circle 1 isn't contained in circle 2. The greatest possible distance between 2 points in the 2 circles is #=d_(C_1C_2)+r_1+r_2=sqrt226+8+9=sqrt226+17~=32.033#

Carson Arnold · Answer

undefined To determine if one circle contains the other, compare the distances between their centers and the difference of their radii. If the distance between the centers of the circles is less than the difference of their radii, then one circle contains the other.

If not, the greatest possible distance between a point on one circle and another point on the other circle is the sum of their radii. This occurs when the centers of the circles are on opposite sides of the line connecting the centers and the line segment connecting the centers is perpendicular to it. Calculate the distances between the centers and add the radii to find this distance.