Question

Two circles have the following equations `(x +2 )^2+(y -5 )^2= 16` and `(x +4 )^2+(y +1 )^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

The two circles intersect each other and larger circle does not contain the smaller circle. The greatest possible distance between a point on one circle and another point on the other is `15.4728`.

Explanation:

Whether the two circles with given centers and radii intersect each other, touch each other (internally or externally) are separated and do not overlap or bigger circle engulfs smaller circle, critically depends on the distance between the centers and sum / difference of their radii . One can find the details here.

Here as the equation of first circle is `(x+2)^2+(y-5)^2=16`, its center is `(-2,5)` and radius is `4`. And as the equation of second circle is `(x+4)^2+(y+1)^2=49` its center is `(-4,-1)` and radius is `7`.

Hence sum of radii is `11` and difference of radii is `3`.

The distance between centers is `sqrt((-4-(-2))^2+(-1-5)^2)=sqrt(4+16)=sqrt20=4.4728`.

As sum of radii is greater than the distance between centers and the difference between radii is smaller than distance between centers, the two circles intersect each other and larger circle does not contain the smaller circle. The greatest possible distance between a point on one circle and another point on the other is `7+4+4.4728=15.4728`.
graph{(x^2+y^2+4x-10y-13)(x^2+y^2+8x+2y-32)=0 [-21.16, 18.84, -8.24, 11.76]}