Question

Two circles have the following equations: `(x -8 )^2+(y -5 )^2= 64` and `(x -7 )^2+(y +2 )^2= 25`. Does one circle contain the other? If not, what is the greatest possible distance between a point on one circle and another point on the other?

Answer 1

The circles overlap.
The greatest distance is `=20.07`

Explanation:

The center of circle A is `c_A=(8,5)` and the radius is `r_A=8`

The center of circle B is `c_b=(7,-2)` and the radius is `r_b=5`

The distance between the centres of the cicles is

`d=sqrt((8-7)^2+(5--2)^2)`

`=sqrt(1+49)=sqrt50`

The sum of the radii is `r_A+r_B=8+5=13`

Therefore,

`d < r_A + r_B`

So, the circles overlap.

The slope of the line joining the centers is `=(5--2)/(8-7)=7`

The greatest possible distance is `d-(max)=r_A+r_B+d`

`=8+5+sqrt50=13+7.07=20.07`

graph{( (x-8)^2) +(y-5)^2-64))((x-7)^2 + (y+2)^2-25)(y-5-7(x-8)) = 0 [-14.26, 14.24, -7.11, 7.11]}