Circle A has a radius of #4 # and a center of #(8 ,5 )#. Circle B has a radius of #2 # and a center of #(6 ,1 )#. If circle B is translated by #<3 ,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 compare the distance (d )"#
#"between the centres of the circles to the"#
#color(blue)"sum of the radii"#
#• " if the sum of radii > d then circles overlap"#
#• " if the sum of the radii < d then no overlap"# Before calculating d we require to find the coordinates of the new centre of B under the given translation which does not change the shape of the circle only its position.
#"under a translation "((3),(1))#
#(6,1)to(6+3,1+1)to(9,2)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)|)))#
#(x_1,y_1)=(8,5),(x_2,y_2)=(9,2)#
#d=sqrt((9-8)^2+(2-5)^2)=sqrt(1+9)=sqrt10~~3.162#
#"sum of radii "=4+2=6#
#"since sum of radii > d then circles overlap"#
graph{(y^2-10y+x^2-16x+73)(y^2-4y+x^2-18x+81)=0 [-20, 20, -10, 10]}
