1.

Prove that the circle `x^(2)+y^(2)+2x+2y+1=0` and circle `x^(2)+y^(2)-4x-6y-3=0` touch each other.

Answer» For first circle `x^(2)+y^(2)+2x+2y+1=0`
`2g_(1)=2,2f_(1)=2,c_(1)=1`
`rArrg_(1)=1,f_(1)=1,c_(1)=1`
`:.` Centre `= P_(1)(-g_(1),-f_(1))=P_(1)(-1,-1)`
and radius `=r_(1)=sqrt(g_(1)^(2)+f_(1)^(2)-c_(1))`
`=sqrt(1+1-1)=1` units
For second circle `x^(2)+y^(2)-4x-6y-3=0`
`2g_(2)=-4,2f_(2)=-6,c_(2)=-3`
`rArrg_(2)=-2,f_(2)=-3,c_(2)=-3`
`:.` Centre `= P_(2)(-g_(2),-f_(2))=P_(2)(2,3)`
and radius `r_(2)=sqrt(g_(2)^(2)+f_(2)^(2)-c_(2))`
`=sqrt(4+9+3)=4` units
Now the distance between the centre of circles
`P_(1)P_(2)=sqrt((2+1)^(2)+(3+1)^(2))`
`=sqrt(9+16)=sqrt(25)=5`
`=r_(1)+r_(2)`
Therefore, two circles touch each other.


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