Circle A has a radius of #1 # and a center of #(2 ,4 )#. Circle B has a radius of #2 # and a center of #(4 ,7 )#. If circle B is translated by #<1 ,-4 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

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"#

Under a translation #((1),(-4))#

#(4,7)to(4+1,7-4)to(5,3)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"#

let # (x_1,y_1)=(2,4)" and " (x_2,y_2)=(5,3)#

#d=sqrt((5-2)^2+(3-4)^2)=sqrt(9+1)=sqrt10≈3.162#

#=3.162-3=0.162" to 3 decimal places"# graph{(y^2-8y+x^2-4x+19)(y^2-6y+x^2-10x+30)=0 [-14.24, 14.24, -7.11, 7.13]}

To determine if circle B overlaps circle A after being translated by <1, -4>, we need to check if the distance between the centers of the two circles plus the radius of circle B is greater than the distance between the two centers. The distance between the centers of the circles A and B before translation is given by the distance formula: \[ \sqrt{(4 - 2)^2 + (7 - 4)^2} = \sqrt{2^2 + 3^2} = \sqrt{13} \] After translating circle B by <1, -4>, the new center of circle B becomes (4 + 1, 7 - 4), which is (5, 3). Now, the distance between the centers of the two circles after translation is: \[ \sqrt{(5 - 2)^2 + (3 - 4)^2} = \sqrt{3^2 + (-1)^2} = \sqrt{10} \] The sum of the radius of circle B and this distance is \( 2 + \sqrt{10} \). So, if \( 2 + \sqrt{10} > \sqrt{13} \), then circle B overlaps circle A. Otherwise, they do not overlap. Calculating, \( 2 + \sqrt{10} \approx 5.16 \) and \( \sqrt{13} \approx 3.61 \). Since \( 5.16 > 3.61 \), circle B overlaps circle A. Therefore, there is no minimum distance between points on both circles because they overlap.