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

Ella Armstrong · Accepted Answer

circles overlap.

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 the radii"# • If sum of radii > d , then circles overlap • If sum of radii < d , then no overlap Before doing this we require to find the new centre of circle B under the given translation, which does not change the shape of the circle, only it's position. Under translation #((2),(1))# #(3,4)to(3+2,4+1)to(5,5)larr" new centre of B"# 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))|)))# where # (x_1,y_1)" and " (x_2,y_2)" are 2 coordinate points"# here the 2 points are (1 ,2) and (5 ,5) the centres of the circles. let #(x_1,y_1)=(1,2)" and " (x_2,y_2)=(5,5)# #d=sqrt((5-1)^2+(5-2)^2)=sqrt(16+9)=sqrt25=5# Sum of radii = radius of A + radius of B = 2 + 5 = 7 Since sum of radii > d , then circles overlap. graph{(y^2-4y+x^2-2x+1)(y^2-10y+x^2-10x+25)=0 [-22.5, 22.5, -11.25, 11.25]}

Noah Atkinson · Answer

undefined No, circle B does not overlap circle A after translation. The minimum distance between points on both circles is $ \sqrt{10} $ units.

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

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