How the following pair of circles are situated in the plane ? Als, find the number of common tangents . `(i) x^(2)+(y-1)^(2)=9` and `(x-1)^(2)+y^(2)=25` (ii) `x^(2)+y^(2)-12x-12y=0` and `x^(2)+y^(2)+6x+6y=0`

Answer

(i) One circle completely lying inside other. No common tangent.
(ii) Touching externally. Three common tangents.

Explanation

(i) Centres of the given circles are `C_(1)(0,1), C_(2)(1,0)` and corresponding radii are `r_(1)=3, r_(2)=5`.
`C_(1)C_(2)=sqrt(2) lt (r_(2)-r_(1))`
Therefore, one circle lies entirely inside the other.
Hence, there is no common tangent.
(ii) Centres of the given circles are `C_(1)(6,6), C_(2)(-3,-3)` and corresponding radii are `r_(1)=6sqrt(2), r_(2)=3sqrt(2)`.
`C_(1)C_(2)=9 sqrt(2)=r_(1)+r_(2)`
There, circles are touching externally.
Hence, there are two common tangents.