Circle A has a radius of #2 # and a center of #(7 ,6 )#. Circle B has a radius of #3 # and a center of #(2 ,3 )#. If circle B is translated by #<-1 ,2 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
1 Answer
no overlap , min. distance ≈1.083 units.
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 of circles
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),(2))#
#(2,3)to(2-1,3+2)to(1,5)larr" 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 (7 ,6) and (1 ,5)
let
# (x_1,y_1)=(7,6)" and " (x_2,y_2)=(1,5)#
#d=sqrt((1-7)^2+(5-6)^2)=sqrt(36+1)=sqrt37≈6.083# sum of radii = radius of A + radius of B = 2 + 3 = 5
Since sum of radii < d, then no overlap of circles
min. distance between points = d - sum of radii
#=6.083-5=1.083#
graph{(y^2-12y+x^2-14x+81)(y^2-10y+x^2-2x+17)=0 [-20, 20, -10, 10]}
