Circle A has a radius of #6 # and a center of #(8 ,5 )#. Circle B has a radius of #3 # and a center of #(6 ,7 )#. 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 #6 # and a center of #(8 ,5 )#.  Circle B has a radius of #3 # and a center of #(6 ,7 )#.  If circle B is translated by #<3 ,1 >#, does it overlap circle A?  If not, what is the minimum distance between points on both circles?

Avery Allen · 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 their radii"#

#• "if sum of radii" > d" then circles overlap"#

#• " if sum of radii" < d" then no overlap"#

Before calculating d, we require to find the 'new' centre of B under the give translation, which does not change the shape of the circle only it's position.

#"under the translation" ((3),(1))#

#(6,7)to(6+3,7+1)to(9,8)larrcolor(red)" 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"#

#"2 points here are " (x_1,y_1)=(8,5),(x_2,y_2)=(9,8)#

#d=sqrt((9-8)^2+(8-5)^2)=sqrt(1+9)=sqrt10~~3.162#

#"sum of radii "=6+3=9#

#"Since sum of radii" > d" then circles overlap"# graph{(y^2-10y+x^2-16x+53)(y^2-16y+x^2-18x+136)=0 [-22.8, 22.81, -11.4, 11.4]}

Wyatt Anderson · Answer

undefined To determine if circle B overlaps circle A after being translated by <3, 1>, we need to calculate the distance between the centers of the two circles after the translation. If this distance is less than the sum of the radii of the circles, then they overlap. If not, we need to find the minimum distance between points on both circles.

The distance between the centers of circle A and B before translation:
$ d = \sqrt{(8 - 6)^2 + (5 - 7)^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{8} \approx 2.83 $

The new coordinates for the center of circle B after translation:
$ (6 + 3, 7 + 1) = (9, 8) $

The distance between the centers of circle A and B after translation:
$ d' = \sqrt{(8 - 9)^2 + (5 - 8)^2} = \sqrt{(-1)^2 + (-3)^2} = \sqrt{10} \approx 3.16 $

The sum of the radii of circles A and B:
$ 6 + 3 = 9 $

Since $ d' > 9 $, the circles do not overlap after the translation.

To find the minimum distance between points on both circles, we subtract the sum of the radii from the distance between the centers:
$ \text{Minimum distance} = d' - (6 + 3) = \sqrt{10} - 9 \approx -5.84 $

The negative result indicates that the circles overlap, and the minimum distance between points on both circles is 0, indicating that they are touching.

Circle A has a radius of #6 # and a center of #(8 ,5 )#. Circle B has a radius of #3 # and a center of #(6 ,7 )#. If circle B is translated by #&lt;3 ,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 #6 # and a center of #(8 ,5 )#. Circle B has a radius of #3 # and a center of #(6 ,7 )#. If circle B is translated by #<3 ,1 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?