Two circles have the following equations #(x -1 )^2+(y -4 )^2= 36 # and #(x +5 )^2+(y -7 )^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?

Question

Two circles have the following equations #(x -1  )^2+(y -4   )^2= 36  # and #(x  +5   )^2+(y -7  )^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?

Noah Atkinson · Accepted Answer

No, the circle: #(x-1)^2+(y-4)^2=36# & #(x+5)^2+(y-7)^2=49# are intersecting each other with a distance #3\sqrt5# between their centers

In general, out of two circles of radii #r_1# & #r_2# & with a distance #d# between their centers, one will be contained by the other if and only if #d<|r_1-r_2|# the greatest possible distance between two circles with radii #r_1# & #r_2# & at a distance #d# between the centers is #=r_1+d+r_2# The circle: #(x-1)^2+(y-4)^2=36# has center #(1, 4)# & radius #r_1=6# and the circle: #(x+5)^2+(y-7)^2=49# has center #(-5, 7)# & radius #r_2=7# hence the distance #d# between the centers #(1, 4)# & #(-5, 7)# of circles is #d=\sqrt{(1-(-5))^2+(4-7)^2}=3\sqrt5# hence, the greatest possible distance between given circles is #=r_1+d+r_2# #=6+3\sqrt5+7# #=13+3\sqrt5#

Mason Adams · Answer

undefined The circles represented by the equations $(x - 1)^2 + (y - 4)^2 = 36$ and $(x + 5)^2 + (y - 7)^2 = 49$ do not contain each other. To find the greatest possible distance between a point on one circle and another point on the other circle, you would compute the sum of their radii and subtract the distance between their centers.