Question

Two circles have the following equations `(x -1 )^2+(y -4 )^2= 36` and `(x +5 )^2+(y -7 )^2= 49`. 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

No, the circle: `(x-1)^2+(y-4)^2=36` & `(x+5)^2+(y-7)^2=49` are intersecting each other with a distance `3\sqrt5` between their centers

Explanation:

In general, out of two circles of radii `r_1` & `r_2` & with a distance `d` between their centers, one will be contained by the other if and only if
`d<|r_1-r_2|`
the greatest possible distance between two circles with radii `r_1` & `r_2` & at a distance `d` between the centers is
`=r_1+d+r_2`
The circle: `(x-1)^2+(y-4)^2=36` has center `(1, 4)` & radius `r_1=6` and the circle: `(x+5)^2+(y-7)^2=49` has center `(-5, 7)` & radius `r_2=7`
hence the distance `d` between the centers `(1, 4)` & `(-5, 7)` of circles is
`d=\sqrt{(1-(-5))^2+(4-7)^2}=3\sqrt5`
hence, the greatest possible distance between given circles is
`=r_1+d+r_2`
`=6+3\sqrt5+7`
`=13+3\sqrt5`