Two circles have the following equations #(x -1 )^2+(y -7 )^2= 25 # and #(x +3 )^2+(y +3 )^2= 9 #. 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 -7   )^2= 25  # and #(x  +3   )^2+(y +3  )^2= 9 #.  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

#"no overlap "~~18.77# #"what we have to do here is compare the distance (d)"# #"between the centres to the sum/difference of the radii"# #• " if difference of radii">d" then one circle inside other"# #• " if sum of radii">d" then circles overlap"# #• " if sum of radii"< d" then no overlap"# #(x-1)^2+(y-7)^2=25," centre"=(1,7), r=5# #(x+3)^2+(y+3)^2=9," centre "=(-3,-3),r=3# #"to calculate d use the "color(blue)"distance formula"# #•color(white)(x)d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)# #"let "(x_1,y_1)=(1,7)" and "(x_2,y_2)=(-3,-3)# #d=sqrt((-3-1)^2+(-3-7)^2)# #color(white)(d)=sqrt(16+100)=sqrt116~~10.77# #"difference of radii "=5-3=2# #"sum of radii "=5+3=8# #"since sum of radii"< d" then no overlap"# #"maximum distance "=d+" sum of radii"# #color(white)(xxxxxxxxxxxxxx)=10.77+8=18.77# graph{((x-1)^2+(y-7)^2-25)((x+3)^2+(y+3)^2-9)=0 [-20, 20, -10, 10]}

Theodore Abad · 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 is the sum of their radii, which is 5 + 3 = 8 units.