Two circles have the following equations #(x -1 )^2+(y -4 )^2= 36 # and #(x +5 )^2+(y +5 )^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?

Question

Two circles have the following equations #(x -1  )^2+(y -4   )^2= 36  # and #(x  +5   )^2+(y +5  )^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?

Caroline Aucoin · Accepted Answer

The circles overlap.

The radius of circle #A# is #r_A=sqrt(36)=6# The radius of circle #B# is #r_B=sqrt(81)=9# The distance between the center #=(1,4)# of circle #A# and the center #(-5,-5)# of circle #B# is #d=sqrt((1+5)^2+(4+5)^2)# #=sqrt(6^2+9^2)# #sqrt(36+81)=sqrt117# #=10.81# The total radii add up to #r_A+r_B=6+9=15# Consequently, As #(r_A+r_B )> d# graph{((x-1)^2+(y-4)^2-36)((x+5)^2+(y+5)^2-81)=0 [-28.87, 28.88, -14.43, 14.43]} shows the overlap of the circles.

Noah Atkinson · Answer

undefined To determine if one circle contains the other, we need to compare the radii of the circles to the distance between their centers.

The center of the first circle is (1, 4) and its radius is √36 = 6 units. 
The center of the second circle is (-5, -5) and its radius is √81 = 9 units.

The distance between the centers of the circles can be calculated using the distance formula: √[(x₂ - x₁)² + (y₂ - y₁)²].

So, the distance between the centers of the circles is √[(-5 - 1)² + (-5 - 4)²] = √[(-6)² + (-9)²] = √(36 + 81) = √117 ≈ 10.82 units.

Since the distance between the centers (10.82 units) is greater than the sum of the radii (6 units + 9 units = 15 units), the circles do not overlap, and one does not contain the other.

To find the greatest possible distance between a point on one circle and another point on the other, we add the radii of the two circles.

So, the greatest possible distance between a point on one circle and another point on the other is 6 units (radius of the first circle) + 9 units (radius of the second circle) = 15 units.