Circle A has a radius of #6 # and a center of #(8 ,5 )#. Circle B has a radius of #3 # and a center of #(6 ,7 )#. 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
May 12, 2017

#"circles overlap"#

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 their 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 give translation, which does not change the shape of the circle only it's position.

#"under the translation" ((3),(1))#

#(6,7)to(6+3,7+1)to(9,8)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"#

#"2 points here are " (x_1,y_1)=(8,5),(x_2,y_2)=(9,8)#

#d=sqrt((9-8)^2+(8-5)^2)=sqrt(1+9)=sqrt10~~3.162#

#"sum of radii "=6+3=9#

#"Since sum of radii" > d" then circles overlap"#
graph{(y^2-10y+x^2-16x+53)(y^2-16y+x^2-18x+136)=0 [-22.8, 22.81, -11.4, 11.4]}