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?

1 Answer
Harish Chandra Rajpoot
Jun 19, 2018

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

Explanation:

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#

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