Circle A has a radius of #3 # and a center at #(1 ,2 )#. Circle B has a radius of #5 # and a center at #(3 ,4 )#. If circle B is translated by #<2 ,1 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
1 Answer
Explanation:
What we have to do here is
#color(blue)"compare "# the distance (d) between the centres to the#color(blue)"sum of radii"#
#• " if "a>d" then circles overlap"#
#• " if "a < d" then no overlap"#
#"before we can calculate d we require to find the new "#
#"centre of B under the given translation"#
#"under the translation "<2,1>#
#(3,4)to(3+2,4+1)to(5,5)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)|)))#
#"let "(x_1,y_1)=(1,2)" and "(x_2,y_2)=(5,5)#
#d=sqrt((5-1)^2+(5-2)^2)=sqrt(16+9)=5#
#"sum of radii "=3+5=8#
#"since sum of radii ">d" then circles overlap"#
graph{((x-1)^2+(y-2)^2-9)((x-5)^2+(y-5)^2-25)=0 [-20, 20, -10, 10]}
