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

James Atkins · Accepted Answer

The two circles are external one another and the greatest possible distance is #sqrt 218-8#

The first circle (A) have the centre in Xca=1 nd Yca=4, and its radius is 3. the second circle (B) has the centre in Xcb=-6 and Ycb=-9. Its radius is 5. the distance between the centres is:

#d=sqrt ((Xca-XCb)^2 +(Yca-YCb)^2# #d=sqrt (49+169)=sqrt 218# the sum of the radii is 3+5 =8 < #sqrt 218# The two circles are external one another and the greatest possible distance is #sqrt 218-8#

Violet Aldrich · 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 calculating the sum of their radii and subtracting the distance between their centers. In this case, the sum of the radii is $3 + 5 = 8$, and the distance between their centers is the distance between the points $(-1, 4)$ and $(-6, -9)$, which can be found using the distance formula to be $ \sqrt{(1-(-6))^2 + (4-(-9))^2} = \sqrt{49 + 169} = \sqrt{218}$. Therefore, the greatest possible distance between a point on one circle and another point on the other is $8 - \sqrt{218}$.