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?

1 Answer

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