Equation of the tangent to the circle, at the point (1, -1), whose centre is the point of intersection of the straight lines x - y = 1 and 2x + y = 3 is:

Point of intersection of lines x - y = 1 and 2x + y = 3 is (4/3, 1/3)
Slope of OP = (1/3 + 1)/(4/3 - 1) = 4/3
Slope of tangent = -3/4
Equation of tangenty + 1 = -3/4(x - 1) ⇒ 4y + 4 = -3x + 3 ⇒ x + 4y + 3 = 0