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Consider the circles x^2+(y-1)^2=9,(x-1)...

Consider the circles `x^2+(y-1)^2=9,(x-1)^2+y^2=25.` They are such that these circles touch each other one of these circles lies entirely inside the other each of these circles lies outside the other they intersect at two points.

Text Solution

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(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.
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