Question

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?

Answer 1

Yes, one circle contains the other.

Explanation:

Circle A, `(x-1)^2+(y-4)^2=64`, center `(1,4)`, radius `r_A=8`
Circle B, `(x+3)^2+(y-4)^2=9`, center `(-3,4)`, 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 `(x1,y1)and(x2,y2)` are `(1,4)`, and `(-3,4)`

`d=sqrt((−3−1)^2+(4-4)^2)=sqrt(16)=4`

2) calculate the sum of the radii `(r_A+r_B)`

Sum of radii =`r_ A + r_B =8+3=11`

3) calculate the difference of the radii `(r_A-r_B)`

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.