Question

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?

Answer 1

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Explanation:

The first circle has center `(-6.1)` and radius 7. The second circle has center `(-4,7)` and radius 9.
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.