Two circles have the following equations: (x +6 )^2+(y -1 )^2= 49 and (x +4 )^2+(y +7 )^2= 81 . 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?

1 Answer
Dec 14, 2016

Still working on it. I wanted to give you something to start with. More soon.

Explanation:

The first circle has center (-6.1) and radius 7. The second circle has center (-4,7) and radius 9.source webmathsource webmath
The line joining both centers has equation

y = ax + b where
7 - 1 = a(-4 - -6)
6 = 2a or a=3

One finds b using

7 = 3(-4) + b
or b=19

The line given by the above equation

intersects the top circle at point P given by
(x+4)^2 + (3x + 19-1)^2 = 49
or 10 x^2 + 8x + 116x + 16 + 18^2 = 49

The other point on the bottom circle can be found in a similar way.