Circle A has a radius of #4 # and a center of #(7 ,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?
1 Answer
circles overlap
Explanation:
What we have to do here is compare the distance (d ) between the centres with the sum and difference 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 new centre of B under the given translation. Note that a translation does not change the shape of the figure , only it's position.
Under a translation of
#((2),(4))# B (1 ,2) → (1+2 ,2+4) → (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 points"# The 2 points here are (3 ,6) and (7 ,3)
#d=sqrt((7-3)^2+(3-6)^2)=sqrt25=5# radius of A + radius of B = 4 + 2 = 6
Since sum of radii > d , then circles overlap
graph{(y^2-6y+x^2-14x+42)(y^2-12y+x^2-6x+41)=0 [-20, 20, -10, 10]}
