Find the length of the tangent drawn from any point on the circle `x^2+y^2+2gx+2fy+c_1=0` to the circle `x^2+y^2+2gx+2fy+c_2=0`

Answer

`sqrt(c_(2)-c_(1))`

Explanation

Let `(x_(1),y_(1))` be any point on the circle
`x^(2)+y^(2)+2gx+2fy+c_(1)=0`
`:. x_(1)^(2)+y_(1)^(2)+2gx_(1)+2fy_(1)+c_(1)=0` (1)
The length of the tangent from `(x_(1),y_(1))` to the circle `x^(2)+y^(2)+2gx+2fy+c_(2)=0` is
`sqrt(x_(1)^(2)+y_(1)^(2)+2gx_(1)+2fy_(1)+c_(2))=sqrt(c_(2)-c_(1))` [Using (1)]