Two circles have the following equations: #(x -1 )^2+(y -4 )^2= 9 # and #(x +6 )^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 -1  )^2+(y -4   )^2= 9  # and #(x  +6   )^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?

Allison Allen · Accepted Answer

The circles partially overlap but the larger does not contain the smaller.

The farthest possible distance will be the sum of the two radii plus the distance between the centers: #10 + sqrt(74)# The radius of the first circle is 3. The radius of the second circle is 7 The distance between the centers: #d = sqrt((-6 - 1)^2 + (9 - 4)^2)# #d = sqrt(74) ~~ 8.6# The circles partially overlap but the larger does not contain the smaller. The farthest possible distance will be the sum of the two radii plus the distance between the centers: #10 + sqrt(74)#

Sadie Allen · Answer

undefined No, one circle does not contain 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. The radii of the circles are $3$ and $7$ units respectively. So, the greatest possible distance between a point on one circle and another point on the other is $3 + 7 = 10$ units.

Maverick Smith · 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 sum of the radii of both circles, then subtracting the distance between their centers. In this case, the sum of the radii is $3 + 7 = 10$, and the distance between the centers is $\sqrt{(1 - (-6))^2 + (4 - 9)^2} = \sqrt{49 + 25} = \sqrt{74}$. Therefore, the greatest possible distance between a point on one circle and another point on the other is $10 - \sqrt{74}$.