Question

Circle A has a radius of `4` and a center of `(7 ,3 )`. Circle B has a radius of `2` and a center of `(1 ,2 )`. If circle B is translated by `<2 ,4 >`, 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 compare the distance (d ) between the centres with the sum and difference of the radii.

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

The first step is to calculate the new centre of B under the given translation. Note that a translation does not change the shape of the figure , only it's position.

Under a translation of `((2),(4))`

B (1 ,2) → (1+2 ,2+4) → (3 ,6)

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

`color(red)(|bar(ul(color(white)(a/a)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(a/a)|)))`
where `(x_1,y_1)" and " (x_2,y_2)" are 2 points"`

The 2 points here are (3 ,6) and (7 ,3)

`d=sqrt((7-3)^2+(3-6)^2)=sqrt25=5`

radius of A + radius of B = 4 + 2 = 6

Since sum of radii > d , then circles overlap
graph{(y^2-6y+x^2-14x+42)(y^2-12y+x^2-6x+41)=0 [-20, 20, -10, 10]}