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`

Options

A. `c_(1)-c`
B. `c-c_(1)`
C. `sqrt(c-c_(1))`
D. `sqrt(c_(1)-c)`

Answer

D

Explanation

Let`(alpha, beta)` be a point on `x^(2)+y^(2)+2gx+2fy+c=0`.
Then, the length of the tangent from `(alpha, beta)` to `x^(2)+y^(2)+2gx+2fy+c_(1)=0` is given by
`l=sqrt(alpha^(2)+beta^(2)+2galpha+2fbeta+c_(1))`
Since `(alpha, beta)` lies on `x^(2)+y^(2)+2gx+2fy+c=0`.
`:. alpha^(2)+beta^(2)+2g alpha+2f beta = -c`
Hence, `l=sqrt(c_(1)-c)`.