Circle A has a radius of 1 1 and a center of (2 ,4 )(2,4). Circle B has a radius of 2 2 and a center of (4 ,9 )(4,9). If circle B is translated by <1 ,-4 ><1,−4>, 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 to thecolor(blue)"sum of the radii"sum of the 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 "before calculating d we require to find the new centre
"of B under the given translation which does not change"of B under the given translation which does not change
"the shape of the circle only it's position"the shape of the circle only it's position
"under a translation " ((1),(-4))
(4,9)to(4+1,9-4)to(5,5)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 points are " (x_1,y_1)=(2,4),(x_2,y_2)=(5,5)
d=sqrt((5-2)^2+(5-4)^2)=sqrt10~~3.162
"sum of radii " =1+2=3
"since sum of radii " < d" then no overlap"
"minimum distance "=d-" sum of radii"
rArr"minimum distance " =3.162-3=0.162
graph{(y^2-8y+x^2-4x+19)(y^2-10y+x^2-10x+46)=0 [-10, 10, -5, 5]}
