Two circles have the following equations (x -1 )^2+(y -4 )^2= 25 and (x +3 )^2+(y +3 )^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?

1 Answer
Apr 5, 2016

The 2 circle overlap but neither are contained within the other...
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Explanation:

Given: Two circles
C_1: (x-1)^2 +(x-4)^2=25
C_2: (x+3)^2 +(x+3)^2=49

Required: Find if C_2 contains C_1 and the greatest distance between them? Check if the circles are contained with one another?

Solution Strategy:
A) Find the center of the circles
B) Use distance formula to find the separation distance of the centers
C) Compare to the sum of the radiuses, to distance of seperation
- If r_(12) lt |r_(C_1)+r_(C_2)| and r_(12) lt |r_(C_2)| Contain
- If r_(12) lt |r_(C_1)+r_(C_2)| circles overlap but no containment

color(red)((A)) The centers of the two circles are:
C_1: O_(c_1)(1,4; 5) this reads center at (1,2) with radius r= 5
C_2: O_(c_1)(-3,-3; 7)

color(blue)((B))
Distance formula between two points C_1(x_1,y_1) and C_2(x_2,y_2) is given by:
r_(12)=sqrt[(x_1-x_2)^2+(y_1-y_2)^2 ]
r_(12)=sqrt[(1-(-3))^2+(4-(-3))^2]=sqrt[(4)^2+(7)^2 ]~~8.06

color(green)((C))
Is r_(12) lt |r_(C_1)+r_(C_2)|;
8.06<(7+5)=12 True but 8.06> 7 so just overlap