# 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?

No, the circle: ${\left(x - 1\right)}^{2} + {\left(y - 4\right)}^{2} = 36$ & ${\left(x + 5\right)}^{2} + {\left(y - 7\right)}^{2} = 49$ are intersecting each other with a distance $3 \setminus \sqrt{5}$ 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: ${\left(x - 1\right)}^{2} + {\left(y - 4\right)}^{2} = 36$ has center $\left(1 , 4\right)$ & radius ${r}_{1} = 6$ and the circle: ${\left(x + 5\right)}^{2} + {\left(y - 7\right)}^{2} = 49$ has center $\left(- 5 , 7\right)$ & radius ${r}_{2} = 7$
hence the distance $d$ between the centers $\left(1 , 4\right)$ & $\left(- 5 , 7\right)$ of circles is
$d = \setminus \sqrt{{\left(1 - \left(- 5\right)\right)}^{2} + {\left(4 - 7\right)}^{2}} = 3 \setminus \sqrt{5}$
hence, the greatest possible distance between given circles is
$= {r}_{1} + d + {r}_{2}$
$= 6 + 3 \setminus \sqrt{5} + 7$
$= 13 + 3 \setminus \sqrt{5}$