Two circles have the following equations: #(x -6 )^2+(y -4 )^2= 64 # and #(x -5 )^2+(y -9 )^2= 49 #. 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 -6  )^2+(y -4   )^2= 64  # and #(x  -5   )^2+(y -9  )^2= 49 #.  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?

Maverick Smith · Accepted Answer

The farthest distance that between to points on the circle is the sum of the distance between the centers and the two radii:

#d_c + r_1 + r_2 = sqrt(26) + 8 + 7 ~~ 20.1#

Here is a graph of the two circles:

The circles overlap but the larger circle does not contain the other. This could have been deduced by the following process.

We assign #r^2# to the constant terms to the right of the equal sign in the given equations and then solve for r:

For circle 1:

#r_1^2 = 64#

#r_1 = 8#

For circle 2:

#r_2^2 = 49#

#r_2 = 7#

Compute the distance, #d_c#, between the two centers:

#d_c = sqrt((6 - 5)^2 + (4 - 9)^2)#

#d_c = sqrt((1)^2 + (-5)^2)#

#d_c = sqrt(26) ~~ 5.1#

If the larger circle contained the smaller, then the distance between the centers would be less than difference between the two radii, 1. This is not the case.

The farthest distance that between two points on the circle is the sum of the distance between the centers and the two radii:

#d_c + r_1 + r_2 = sqrt(26) + 8 + 7 ~~ 20.1#

Evelyn Arboleda · Answer

undefined To determine if one circle contains the other, we need to compare their radii and distances between their centers.

The first circle has a center at (6, 4) and a radius of √64 = 8 units. 
The second circle has a center at (5, 9) and a radius of √49 = 7 units.

To see if one circle contains the other, we need to check if the distance between their centers is less than the difference between their radii.

The distance between the centers of the circles can be found using the distance formula: 
√((x2 - x1)^2 + (y2 - y1)^2)

Distance = √((5 - 6)^2 + (9 - 4)^2) = √(1 + 25) = √26

The difference between their radii is |8 - 7| = 1 unit.

Since the distance between their centers (√26) is greater than the difference between their radii (1), neither circle contains the other.

The greatest possible distance between a point on one circle and another point on the other can be found by adding the radii of the circles and then subtracting the distance between their centers.

Maximum distance = 8 + 7 - √26 ≈ 15.82 units.