Two circles have the following equations: #(x -8 )^2+(y -5 )^2= 64 # and #(x -7 )^2+(y +2 )^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 · 2016-12-22T17:54:27

The circles overlap.
The greatest distance is #=20.07#

The center of circle A is #c_A=(8,5)# and the radius is #r_A=8#

The center of circle B is #c_b=(7,-2)# and the radius is #r_b=5#

The distance between the centres of the cicles is

#d=sqrt((8-7)^2+(5--2)^2)#

#=sqrt(1+49)=sqrt50#

The sum of the radii is #r_A+r_B=8+5=13#

Therefore,

#d < r_A + r_B#

So, the circles overlap.

The slope of the line joining the centers is #=(5--2)/(8-7)=7#

The greatest possible distance is #d-(max)=r_A+r_B+d#

#=8+5+sqrt50=13+7.07=20.07#

graph{( (x-8)^2) +(y-5)^2-64))((x-7)^2 + (y+2)^2-25)(y-5-7(x-8)) = 0 [-14.26, 14.24, -7.11, 7.11]}