Circle A has a radius of #4 # and a center of #(6 ,2 )#. Circle B has a radius of #2 # and a center of #(5 ,7 )#. If circle B is translated by ##, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Question

Circle A has a radius of #4 # and a center of #(6 ,2 )#.  Circle B has a radius of #2 # and a center of #(5 ,7 )#.  If circle B is translated by #<-2 ,2 >#, does it overlap circle A?  If not, what is the minimum distance between points on both circles?

Sadie Allen · Accepted Answer

#"no overlap ",~~1.616# What we have to do here is #color(blue)"compare"# the distance ( d )between the centres of the circles to the #color(blue)"sum of radii"# #• " If sum of radii > d, then circles overlap"# #• " If sum of radii < d, then no overlap"# #"Before calculating d we require to find the 'new' centre of B"# #"under the given translation"# which does not change the shape of the circle only it's position. #"Under a translation " ((-2),(2))# #(5,7)to(5-2,7+2)to(3,9)larrcolor(red)" new centre of B"# #"to calculate d use the "color(blue)"distance formula"# #color(red)(bar(ul(|color(white)(2/2)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(2/2)|)))# #"where " (x_1,y_1),(x_2,y_2)" are 2 coordinate points"# #"the 2 points here are " (6,2)" and " (3,9)# #d=sqrt((3-6)^2+(9-2)^2)=sqrt(9+49)=sqrt58~~7.616# #"sum of radii " =4+2=6# #"Since sum of radii"< d" then no overlap"# #"min. distance between points " =d-"sum of radii"# #rArr"min distance "=7.616-6=1.616# graph{(y^2-4y+x^2-12x+24)(y^2-18y+x^2-6x+86)=0 [-28.87, 28.86, -14.43, 14.44]}

Circle A has a radius of #4 # and a center of #(6 ,2 )#. Circle B has a radius of #2 # and a center of #(5 ,7 )#. If circle B is translated by #&lt;-2 ,2 &gt;#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Circle A has a radius of #4 # and a center of #(6 ,2 )#. Circle B has a radius of #2 # and a center of #(5 ,7 )#. If circle B is translated by #<-2 ,2 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?