Question

Two circles have the following equations: `(x +6 )^2+(y -5 )^2= 49` and `(x -9 )^2+(y +4 )^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

Circles do not contain each other, `16+\sqrt306`

Explanation:

The equation of first circle:

`(x+6)^2+(y-5)^2=49`

The above circle has center `(-6, 5)` & radius `r_1=7`

The equation of second circle:

`(x-9)^2+(y+4)^2=81`

The above circle has center `(9, -4)` & radius `r_2=9`

Now, the distance `d` between the centers `(-6, 5)` & `(9, -4)` of circles, is given by distance formula as follows

`d=\sqrt{(-6-9)^2+(5-(-4))^2}`

`=\sqrt{306}`

`=17.493`

`17.493>7+9=16`

`\implies d>r_1+r_2`

since the distance between the centers of circles is greater than the sum of radii hence the circles are externally separated i.e. they do not contain one another or intersect each other

Now, the maximum distance between the points on two given circles will be equal to the distance between the farthest points on two circles which lie on the line joining the centers, hence the maximum possible distance

`=r_1+r_2+d`

`=7+9+\sqrt306`

`=16+\sqrt306`