Circle A has a radius of 6 6 and a center of (8 ,5 )(8,5). Circle B has a radius of 3 3 and a center of (6 ,7 )(6,7). 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
color(blue)"compare"compare the distance ( d ) between the centres of the circles to thecolor(blue)"sum of their radii"sum of their radii
• "if sum of radii" > d" then circles overlap"∙if sum of radii>d then circles overlap
• " if sum of radii" < d" then no 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]}
