Two circles have the following equations: #(x +3 )^2+(y -1 )^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?

1 Answer
Tamir E. · Shwetank Mauria
Nov 16, 2017

One circle does not contain other. Circles intersect each other. Greatest possible distance between point on one circle and another point on the other is #23.44#

Explanation:

By general:
#(x-x_0)^2+(y-y_0)^2=R^2#

#=> center: (-x_0,-y_0)#
#=> R: sqrt(R^2)#

First let's find the centers and the R of each circle:

For #(x+3)^2+(y-1)^2=64#
Center: #(-3,1)#
#R_1#: #sqrt64=8#

For #(x-7)^2+(y+2)^2=25#
Center: #(7,-2)#
#R_2#: #sqrt25=5#

The distance between two points given by Protagoras:
#d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)#
#=>#
The distance between the centers of the circles above given by Protagoras:
#d=sqrt((7-(-3))^2+((-2)-1)^2)=sqrt109~~10.44#

#R_1+R_2=8+5=13# and #R_1-R_2=8-5=3#

We saw that #R_1+R_2>d# and #R_1-R_2 < d# therefore circles intersect each other. Greatest possible distance between point on one circle and another point on the other is #8+5+10.44=23.44#

For details see here.

graph{((x+3)^2+(y-1)^2-64)((x-7)^2+(y+2)^2-25)=0 [-20, 20, -10, 10]}

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