Two circles have the following equations #(x +5 )^2+(y +3 )^2= 9 # and #(x +4 )^2+(y -1 )^2= 1 #. 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
CW
Sep 24, 2016

No overlap of the circles. Greatest distance #=8.123#

Explanation:

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Circle A, center (-5,-3), #r_A=3#
Circle B, center (-4,1), #r_B=1#

Now compare the distance #d# between the centres of the circles to the sum of the radii.

1) If the sum of the radii #>#d, the circles overlap.
2) If the sum of the radii #<#d, then no overlap.
3)If the difference of the radii #>d#, then one circle inside the other.

To calculate #d#, use the distance formula :
#d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)#

where #(x_1,y_1) and (x_2,y_2)# are 2 coordinate points

here the two points are #(-5,-3)# and #(-4,1)# the centres of the circles

let#(x_1,y_1)=(-5,-3)# and #(x_2,y_2)=(-4,1)#

#d=sqrt((-4-(-5))^2+(1-(-3))^2)=sqrt(1^2+(4)^2)=sqrt17=4.123#

Sum of radii = radius of A + radius of B #= 3+1=4#

Since sum of radius #<#d, then no overlap of the circles

greatest distance :
#d+#sum of radii #= 4.123+3+1 =8.123#

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