Two circles have the following equations: #(x +2 )^2+(y -1 )^2= 49 # and #(x +4 )^2+(y +7 )^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?

Question

Two circles have the following equations: #(x +2  )^2+(y -1   )^2= 49  # and #(x  +4   )^2+(y +7  )^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?

Mason Adams · Accepted Answer

The circles overlap and the greatest distance is #=20.25#

The radius of the circles are

#r_A=7#

#r_B=5#

The sum of the radii is

#r_A+r_B=7+5=12#

The centers of the circles are #(-2,1)# and #(-4,-7)#

The distance between the centers is

#d=sqrt((-4-(-2))^2+(-7-(1))^2)#

#=sqrt((-2)^2+(-8)^2)#

#sqrt(4+64)=sqrt68=8.25#

As,

#(r_A+r_B) >d#, the circles overlap

The greatest distance is

#=d+r_A+r_B=12+8.25=20.25#

Parker Allen · Answer

undefined To determine whether one circle contains the other, we can compare their radii. The first circle has a radius of √49 = 7, while the second circle has a radius of √25 = 5. Since the radius of the first circle is greater than the radius of the second circle, the second circle cannot contain the first circle.

To find the greatest possible distance between a point on one circle and another point on the other, we need to find the distance between their centers and subtract the sum of their radii.

The center of the first circle is (-2, 1), and the center of the second circle is (-4, -7).

Using the distance formula, the distance between the centers is:

√[(-4 - (-2))^2 + (-7 - 1)^2] = √[(-2)^2 + (-8)^2] = √(4 + 64) = √68.

The sum of the radii is 7 + 5 = 12.

So, the greatest possible distance between a point on one circle and another point on the other is √68 - 12.