# Two circles have the following equations (x -4 )^2+(y +3 )^2= 9  and (x +4 )^2+(y -1 )^2= 16 . 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?

Sep 6, 2016

Circles are outside each other and greatest possible distance between a point on one circle and another point on the other is $14.2111$.

#### Explanation:

The center of circle ${\left(x - 4\right)}^{2} + {\left(y + 3\right)}^{2} = 9$ is $\left(2 , - 3\right)$ and radius is $3$ and center of circle ${\left(x + 4\right)}^{2} + {\left(y - 1\right)}^{2} = 16$ is $\left(- 4 , 1\right)$ and radius is $4$.

The distance between centers is sqrt((-4-2)^2+(1-(-3))^2

= $\sqrt{36 + 16} = \sqrt{52} = 7.2111$

If the radii of two circles is ${r}_{1}$ and ${r}_{2}$ and we also assume that ${r}_{1} > {r}_{2}$ and the distance between centers is $d$, then

A - if ${r}_{1} + {r}_{2} = d$, they touch each other externally and greatest possible distance between a point on one circle and another point on the other is $2 \left({r}_{1} + {r}_{2}\right)$ and smallest possible distance between a point on one circle and another point on the other is $0$.

B - if ${r}_{1} + {r}_{2} < d$, they do not touch each other and are outside each other (i.e. one is not contained in other) and greatest possible distance between a point on one circle and another point on the other is ${r}_{1} + {r}_{2} + d$ and smallest possible distance between a point on one circle and another point on the other is $d - {r}_{1} - {r}_{2}$.

C - if ${r}_{1} + {r}_{2} > d$ and ${r}_{1} - {r}_{2} = d$, they touch each other internally and smaller circle is contained in other and greatest possible distance between a point on one circle and another point on the other is $2 {r}_{2}$ and smallest distance is $0$.

D - if ${r}_{1} + {r}_{2} > d$ and ${r}_{1} - {r}_{2} > d$, smaller circle lies inside the larger circle and greatest possible distance between a point on one circle and another point on the other is ${r}_{1} + {r}_{2} + d$ and smallest distance is ${r}_{1} - {r}_{2} - d$.

E - if ${r}_{1} + {r}_{2} > d$ and ${r}_{1} - {r}_{2} < d$, the two circles intersect each other and greatest possible distance between a point on one circle and another point on the other is ${r}_{1} + {r}_{2} + d$ and smallest distance is $0$.

Now, here as ${r}_{1} + {r}_{2} = 7 < d = 7.2111$ and ${r}_{1} - {r}_{2} < d$, they do not touch each other and are outside each other (i.e. one is not contained in other) and greatest possible distance between a point on one circle and another point on the other is $4 + 3 + 7.2111 = 14.2111$.

graph{(x^2+y^2-8x+6y+16)(x^2+y^2+8x-2y+1)=0 [-10.5, 9.5, -5.665, 4.755]}