Two circles have the following equations: #(x +3 )^2+(y -5 )^2= 64 # and #(x -2 )^2+(y +4 )^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 +3  )^2+(y -5   )^2= 64  # and #(x  -2   )^2+(y +4  )^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?

Noah Atkinson · Accepted Answer

No. One circle does not contain the other.

Faced with this problem what can you do? Sketch it One circle has centre (-3,5)and radius 8 The other has centre (2,-4) and radius 9 We know this by inspection of the equations.. So draw that line through both centres and out to the circumferences. How far apart are the centres? Difference in x coordinates 5, difference in y coordinates 9, from Pythagoras answer#sqrt106# As this is a little over 10 clearly even the centre of the smaller circle is not within the larger. What is the furthest distance of one circumference from the other? 8 +9+#sqrt106#

Autumn Armstrong · Answer

undefined The 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 adding the radii of the circles and subtracting the distance between their centers. The radii are √64 = 8 and √81 = 9. The distance between the centers of the circles can be calculated using the distance formula. The centers of the circles are (-3, 5) and (2, -4). Therefore, the distance between the centers is √((2 - (-3))^2 + (-4 - 5)^2) = √(5^2 + 9^2) = √(25 + 81) = √106. So, the greatest possible distance between a point on one circle and another point on the other is 8 + 9 - √106.