Two circles have the following equations (x -1 )^2+(y -4 )^2= 64  and (x +3 )^2+(y +1 )^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 28, 2017

See below.

Explanation:

We can determine whether two circle intersect at two points, touch at one point only or do not touch.

If the sum of their radii is less than the distance of their centres the the circles do not touch. If the sum of their radii is greater than the distance between their centres then the circles intersect at two points. If the sum of their radii is equal to the distance between their centres then the circles touch at one point

First we need to find the distance between there centres. We can do this with the distance formula.

Distance formula is:

$d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

$d = \sqrt{{\left(1 - \left(- 3\right)\right)}^{2} + {\left(4 - \left(- 1\right)\right)}^{2}} \implies \sqrt{16 + 25} = \sqrt{41}$

sum = $\sqrt{9} + \sqrt{64} \implies 3 + 8 = 11$
$11 > \sqrt{41}$, so they intersect at two points.
Greatest distance = $\sqrt{41} + 3 + 8 = 17.032$