Two circles have the following equations: #(x +6 )^2+(y -5 )^2= 64 # and #(x -2 )^2+(y +4 )^2= 81 #. 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
Shwetank Mauria
Sep 7, 2016

Circles intersect each other and greatest possible distance between a point on one circle and another point on the other is #29.0416#.

Explanation:

Please refer to details here.

The center of circle #(x+6)^2+(y-5)^2=64# is #(-6,5)# and radius is #8# and center of circle #(x-2)^2+(y+4)^2=81# is #(2,-4)# and radius is #9#.

The distance between centers is #sqrt((2-(-6))^2+(-4-5)^2#

= #sqrt(64+81)=sqrt145=12.0416#

Let the radii of two circles is #r_1# and #r_2# and we also assume that #r_1>r_2# and the distance between centers is #d#.

Then as #r_1+r_2=17 > d=12.0416# and #r_1-r_2=1 < d=12.0416#, the two circles intersect each other and greatest possible distance between a point on one circle and another point on the other is #9+8+12.0416=29.0416#
graph{(x^2+y^2+12x-10y-3)(x^2+y^2-4x+8y-61)=0 [-40, 40, -20, 20]}

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