Question

Circle A has a radius of `2` and a center of `(2 ,6 )`. Circle B has a radius of `3` and a center of `(7 ,8 )`. If circle B is translated by `<-2 ,-3 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

`"circles overlap"`

Explanation:

.What we have to do here is `color(blue)"compare "`the distance (d) between the centres to the `color(blue)"sum of the radii"`

`• " if sum of radii"> d" then circles overlap"`

`• " if sum of radii "< d" then no overlap"`

`"Before calculating d we require to find the 'new centre'"`
`"of B under the given translation"`

`(7,8)to(7-2,8-3)to(5,5)larrcolor(red)"new centre of B"`

`"to calculate d use the "color(blue)"distance formula"`

`d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`

`"let "(x_1,y_1)=(5,5)" and "(x_2,y_2)=(2,6)`

`d=sqrt((2-5)^2+(6-5)^2)=sqrt(9+1)=sqrt10~~3.16`

`"sum of radii "=2+3=5`

`"Since sum of radii ">d" then circles overlap"`
graph{((x-2)^2+(y-6)^2-4)((x-5)^2+(y-5)^2-9)=0 [-20, 20, -10, 10]}