The lengths of the tangents from `P(1,-1)` and `Q(3,3)` to a circle are `sqrt(2)` and `sqrt(6)` , respectively. Then, find the length of the tangent from `R(-1,-5)` to the same circle.

Explanation

Let the circle be `x^(2)+y^(2)+2gx+2fy+c=0`.
Let the tangents from points P,Q,R touch the circle at A,B,C, respectively. According to question,
`PA^(2)=2`
or `1+1+2g-2f+c=2`
or `2g-2f+c=0` (1)
and `PB^(2)=5`
or `9+9+6g+6f+c=6`
or `6g+6f+c= -12` (2)
From (1) and (2),
`g= -1-(c)/(3),f=-1+(c)/(6)`
Now, `RC^(2)=1+25-2g-10f+c`
`=26+2+(2c)/(3)+10-(5c)/(3)+c`
`=38`
`:. RC =sqrt(38)`