Circle A has a radius of #4 # and a center of #(5 ,3 )#. Circle B has a radius of #2 # and a center of #(1 ,2 )#. If circle B is translated by ##, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Question

Circle A has a radius of #4 # and a center of #(5 ,3 )#.  Circle B has a radius of #2 # and a center of #(1 ,2 )#.  If circle B is translated by #<2 ,4 >#, does it overlap circle A?  If not, what is the minimum distance between points on both circles?

Sadie Adams · Accepted Answer

circles overlap.

What we have to do here is compare the distance ( d) between the centres of the circles with the sum of the radii. • If sum of radii > d , then circles overlap • If sum of radii < d , then no overlap The first step is to calculate the coordinates of the 'new' centre of circle B under the given translation. Under a translation the circle remains a circle but it's position changes. Under a translation #((2),(4))# (1 ,2) → (1+2 ,2+4) → new centre of B is (3 ,6) To calculate d, use the #color(blue)"distance formula"# #color(red)(|bar(ul(color(white)(a/a)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(a/a)|)))# where # (x_1,y_1)" and " (x_2,y_2)" are 2 coordinate points"# Here the 2 points are (5 ,3) and (3 ,6) the centres of the circles. let #(x_1,y_1)=(5,3)" and " (x_2,y_2)=(3,6)# #d=sqrt((3-5)^2+(6-3)^2)=sqrt(4+9)=sqrt13≈3.606# sum of radii = radius of A + radius of B = 4 + 2 = 6 Since sum of radii > d , then circles overlap graph{(y^2-6y+x^2-10x+18)(y^2-12y+x^2-6x+41)=0 [-20, 20, -10, 10]}

Circle A has a radius of #4 # and a center of #(5 ,3 )#. Circle B has a radius of #2 # and a center of #(1 ,2 )#. If circle B is translated by #&lt;2 ,4 &gt;#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Circle A has a radius of #4 # and a center of #(5 ,3 )#. Circle B has a radius of #2 # and a center of #(1 ,2 )#. If circle B is translated by #<2 ,4 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?