Two circles have the following equations #(x -1 )^2+(y -4 )^2= 36 # and #(x +5 )^2+(y -2 )^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 · 2018-04-16T13:10:02

No , they will intersect at two points.Greatest possible distance between two points of two circles is #19.68# unit

Center of first circle #(x-1)^2+(y-4)^2=6^2# is #(1,4)#

and radius is #6# unit .

Center of second circle #(x+5)^2+(y-2)^2=7^2# is #(-5,2)#

and radius is #7# unit . Distance between their centers is

#d=sqrt((x_1-x_2)^2+(y_1-y_2)^2)=sqrt((1+5)^2+(4-2)^2) # or

#d=sqrt 40 ~~ 6.32 # unit.

Sum of their radii is # r_1+r_2= 6+7=13# unit

Difference of their radii is # r_2-r_1= 7-6=1# unit

Here # (r_2-r_1) < d < (r_1+r_2) or 1<6.32<13 # , so the circles

will intersect at two points. So one circle will not contain the

other. Condition for containing one on other is #d<=|r_1-r_2|#

Greatest possible distance between two points is #2(r_1+r_2)-d#

#12+14-6.32 ~~ 19.68# unit [Ans]