Prove that the radii of the circles `x^(2)+y^(2)=1,x^(2)=y^(2)-2x-6y=6andx^(2)+y^(2)-4x-12y=9` are in AP.

Explanation

For the circle `x^(2)+y^(2)=1`
Radius `r_(1)=sqrt(1)=1` unit
For the circle `x^(2)+y^(2)-2x-6y=6`
`x^(2)+y^(2)-2x-6y-6=0`
`:.2g_(2)=-2.2f_(2)=-6,c_(2)=-6`
`rArr g_(2)=-1,f_(2)=-3,c_(2)=-6`
and radius `r_(2)=sqrt(g_(2)^(2)+f_(2)^(2)-c)`
`=sqrt(1+9+6)=sqrt(16)=4` unit
For the circle `x^(2)+y^(2)-4x-12y-9=0`
`2g_(3)=-4,2f_(3)=-12,c_(3)=-9`
`rArrg_(3)=-2,f_(3)=-6,c_(3)=-9`
`:. r_(3)=sqrt(g_(3)^(2)+f_(3)^(2)-c_(3))`
`= sqrt(4+36+9)=sqrt(49)=7` units
Now `r_(1)+r_(3)=1+7=8=2xx4`
`=2r_(2)`
`:. r_(1),r_(2),r_(3)` are in A.P.