# Two circles have the following equations (x -1 )^2+(y -4 )^2= 64  and (x +3 )^2+(y -4 )^2= 9 . 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 26, 2016

Yes, one circle contains the other.

#### Explanation:

Circle A, ${\left(x - 1\right)}^{2} + {\left(y - 4\right)}^{2} = 64$, center $\left(1 , 4\right)$, radius ${r}_{A} = 8$
Circle B, ${\left(x + 3\right)}^{2} + {\left(y - 4\right)}^{2} = 9$, center $\left(- 3 , 4\right)$, radius ${r}_{B} = 3$

1) calculate d the distance between the centres of the circles, use the distance formula :
d=sqrt((x2−x1)^2+(y2−y1))^2

where $\left(x 1 , y 1\right) \mathmr{and} \left(x 2 , y 2\right)$ are $\left(1 , 4\right)$, and $\left(- 3 , 4\right)$

d=sqrt((−3−1)^2+(4-4)^2)=sqrt(16)=4

2) calculate the sum of the radii $\left({r}_{A} + {r}_{B}\right)$

Sum of radii =${r}_{A} + {r}_{B} = 8 + 3 = 11$

3) calculate the difference of the radii $\left({r}_{A} - {r}_{B}\right)$

Difference of Radii ${r}_{A} - {r}_{B} = 8 - 3 = 5$

3 )compare the distance d between the centres of the circles to the sum of the radii and to the difference of the radii.

1) If ${r}_{A} + {r}_{B} > d$, the circles overlap.
2) If ${r}_{A} + {r}_{B} < d$, then no overlap.
3) If ${r}_{A} - {r}_{B} > d$, then Circle A contain Circle B.

In our case, since ${r}_{A} - {r}_{B} > d$,
Hence, Circle B is contained in Circle A.