Circle A has a radius of #2 # and a center of #(6 ,6 )#. Circle B has a radius of #3 # and a center of #(2 ,4 )#. If circle B is translated by #<1 ,5 >#, 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 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 the translation "((1),(5))#
#(2,4)to(2+1,4+5)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)|)))#
#"let "(x_1,y_1)=(6,6)" and "(x_2,y_2)=(3,9)#
#d=sqrt((3-6)^2+(9-6)^2)=sqrt18~~ 4.243#
#"sum of radii "=2+3=5#
#"since sum of radii">d" then circles overlap"#
graph{((x-6)^2+(y-6)^2-4)((x-3)^2+(y-9)^2-9)=0 [-20, 20, -10, 10]}
