Circle A has a radius of #3 # and a center at #(1 ,3 )#. Circle B has a radius of #5 # and a center at #(1 ,7 )#. 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?
1 Answer
no overlap , ≈0.246
Explanation:
What we have to do here is compare the distance (d) between the centres to the sum of the radii.
• If sum of radii > d , then circles overlap
• If sum of radii < d , then no overlap
Our first step is to find the new centre of B under the given translation.The shape of a figure does not change under a translation only it's position.
Under a translation
#((2),(4))# centre of B (1 ,7) → (1+2 ,7+4) → (3 ,11)
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 points"# Here the 2 points are (1 ,3) and (3 ,11)
#d=sqrt((3-1)^2+(11-3)^2)=sqrt68≈8.246# radius of A + radius of B = 3 + 5 = 8
Since sum of radii < d , then no overlap.
minimum distance between 2 points = 8.246 - 8 = 0.246
graph{(y^2-6y+x^2-2x+1)(y^2-22y+x^2-6x+105)=0 [-20, 20, -10, 10]}
