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.

The lengths of the tangents from `P(1,-1)`and `Q(3,3)`to a circle are

1.	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.
Answer» 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)`

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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.

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