Circle A has a radius of #2 # and a center of #(7 ,2 )#. Circle B has a radius of #3 # and a center of #(5 ,7 )#. If circle B is translated by #<-1 ,2 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

no overlap , ≈ 2.616 units.

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

Under a translation #((-1),(2))#
#B(5,7)to(5-1,7+2)to(4,9)larr" 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)" and " (x_2,y_2)" are 2 coordinate points"#

The 2 points here are (7 ,2) and (4 ,9)

let # (x_1,y_1)=(7,2)" and " (x_2,y_2)=(4,9)#
#d=sqrt((4-7)^2+(9-2)^2)=sqrt(9+49)=sqrt58≈7.616#

Sum of radii = radius of A + radius of B = 2 + 3 = 5

Since sum of radii < d, then no overlap

and min. distance between points = d - sum of radii

#=7.616-5=2.616# graph{(y^2-4y+x^2-14x+49)(y^2-18y+x^2-8x+88)=0 [-25.31, 25.32, -12.66, 12.65]}
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Answer 2

To determine if circle B overlaps circle A after translation, calculate the distance between the centers of the circles. If this distance is less than the sum of the radii of both circles, they overlap. If not, the minimum distance between points on both circles is the difference between the distance between their centers and the sum of their radii.

Let ( C_A = (7, 2) ) be the center of circle A, and ( C_B = (5, 7) ) be the center of circle B before translation.

The distance between the centers of the circles is: [ d = \sqrt{(5 - 7)^2 + (7 - 2)^2} = \sqrt{4 + 25} = \sqrt{29} ]

The sum of the radii of both circles is: [ r_A + r_B = 2 + 3 = 5 ]

The translated center of circle B is ( C_B' = (5 - 1, 7 + 2) = (4, 9) ).

The distance between the centers of the translated circle B and circle A is: [ d' = \sqrt{(7 - 4)^2 + (2 - 9)^2} = \sqrt{9 + 49} = \sqrt{58} ]

Since ( \sqrt{58} > 5 ), the circles do not overlap after translation.

The minimum distance between points on both circles is the absolute difference between the distance between their centers and the sum of their radii: [ \text{Minimum distance} = \sqrt{58} - 5 ]

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Answer from HIX Tutor

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