The length of tangents from P(1, -1) and Q(3,3) to a circle are $\sqrt{2}and \sqrt{6}$ respectively, then the length of tangent from R(-2,-7) to the same circle is

Length of tangent from (x1, y1) to circle x^2 + y^2 + 2gx + 2fy + c = 0 is sqrt(x1^2 + y1^2 + 2gx1 + 2fy1 + c). Let the circle be x^2 + y^2 + 2gx + 2fy + c = 0. Tangent from P(1,-1) is sqrt(1+1+2g-2f+c) = sqrt(2), so 2+2g-2f+c = 2 => 2g-2f+c = 0. Tangent from Q(3,3) is sqrt(9+9+6g+6f+c) = sqrt(6), so 18+6g+6f+c = 6 => 6g+6f+c = -12. Tangent from R(-2,-7) is sqrt(4+49-4g-14f+c) = sqrt(53-4g-14f+c). Solving the system for g, f, c is underdetermined, but the power of the point can be found.