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?

Question

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?

Avery Allen · Accepted Answer

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

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#

James Atkins · Answer

undefined The two circles do not contain each other. The greatest possible distance between a point on one circle and another point on the other can be found by calculating the distance between their centers and then adding their radii. The distance between the centers of the circles can be found using the distance formula. Let's denote the centers of the circles as $(x_1, y_1)$ and $(x_2, y_2)$. Then the distance between their centers is given by:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Once you have the distance between the centers, add the radius of one circle to the distance to find the greatest possible distance between a point on one circle and another point on the other.