Circle A has a radius of 4 4 and a center of (6 ,2 )(6,2). Circle B has a radius of 2 2 and a center of (5 ,7 )(5,7). If circle B is translated by <-2 ,2 ><−2,2>, 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 radii"sum of 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"Before calculating d we require to find the 'new' centre of B
"under the given translation"under the given translation which does not change the shape of the circle only it's position.
"Under a translation " ((-2),(2))
(5,7)to(5-2,7+2)to(3,9)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"
"the 2 points here are " (6,2)" and " (3,9)
d=sqrt((3-6)^2+(9-2)^2)=sqrt(9+49)=sqrt58~~7.616
"sum of radii " =4+2=6
"Since sum of radii"< d" then no overlap"
"min. distance between points " =d-"sum of radii"
rArr"min distance "=7.616-6=1.616
graph{(y^2-4y+x^2-12x+24)(y^2-18y+x^2-6x+86)=0 [-28.87, 28.86, -14.43, 14.44]}
