Question

Circle A has a radius of `5` and a center of `(2 ,7 )`. Circle B has a radius of `1` and a center of `(3 ,1 )`. If circle B is translated by `<1 ,3 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

They do not overlap and the minimum distance between the two circles is >1.

Explanation:

To figure this out, the simplest thing to do is to graph it out. From all the information given, we can find the equations for both circles and graph them. A circle's equation is:

`x^2+y^2=r^2`

`r` in this case equals the `radius` of the circle.

Also, the problem gives us the center point at which the circle sits. This will also help with forming the equations.

Circle A is `(x-2)^2 + (y-7)^2 = 25`
Circle B is `(x-3)^2 + (y-1)^2 = 1`

Now that we have our equations, we can graph them. graph{(x-2)^2+(y-7)^2=25 [-13.39, 14.7, 0.26, 14.31]}

graph{(x-3)^2+(y-1)^2=1 [-11.13, 13.84, -1.94, 10.55]}

Now, let's visualize Circle B translated to the right one, and up three. This will make Circle B's new equation look like:

`(x-4)^2 + (y-10)^2 = 1`
graph{(x-4)^2+(y-10)^2=1 [-6.25, 13.48, 3.71, 13.58]}

If we now compare the graph of Circle A with the translated graph of Circle B, we can see that they still do not overlap and the minimum distance between the two circles is >1.

Answer 2

`"circle B is inside circle A"`

Explanation:

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

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

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

`• " if difference of radii">d" then 1 circle inside other"`

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

`"under a translation "<1,3>`

`(3,1)to(3+1,1+3)to(4,4)larrcolor(red)"new centre of B"`

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

`•color(white)(x)d=sqrt((x_2-x_1)^2+(y_2-y_1)^2`

`"let "(x_1,y_1)=2,7)" and "(x_2,y_2)=(4,4)`

`d=sqrt((4-2)^2+(4-7)^2)=sqrt(4+9)=sqrt13~~3.61`

`"sum of radii "=5+1=6`

`"difference of radii "=5-1=4`

`"Since diff. of radii">d" then 1 circle inside other"`
graph{((x-2)^2+(y-7)^2-25)((x-4)^2+(y-4)^2-1)=0 [-40, 40, -20, 20]}