Circle A has a radius of 4 4 and a center of (8 ,5 )(8,5). Circle B has a radius of 2 2 and a center of (6 ,1 )(6,1). If circle B is translated by <3 ,1 ><3,1>, does it overlap circle A? If not, what is the minimum distance between points on both circles?
1 Answer
Explanation:
"what we have to do here is compare the distance (d )"what we have to do here is compare the distance (d )
"between the centres of the circles to the"between the centres of the circles to the
color(blue)"sum of the radii"sum of the radii
• " if the sum of radii > d then circles overlap"∙ if the sum of radii > d then circles overlap
• " if the sum of the radii < d then no overlap"∙ if the sum of the radii < d then no overlap Before calculating d we require to find the coordinates of the new centre of B under the given translation which does not change the shape of the circle only its position.
"under a translation "((3),(1))
(6,1)to(6+3,1+1)to(9,2)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)|)))
(x_1,y_1)=(8,5),(x_2,y_2)=(9,2)
d=sqrt((9-8)^2+(2-5)^2)=sqrt(1+9)=sqrt10~~3.162
"sum of radii "=4+2=6
"since sum of radii > d then circles overlap"
graph{(y^2-10y+x^2-16x+73)(y^2-4y+x^2-18x+81)=0 [-20, 20, -10, 10]}
