A circle passes through the point (0,1) and touches the parabola y = x^2 at the point (1,1) . The centre of the circle is...

Step-by-step solution

The equation of the tangent to the parabola y = x^2 at the point (1,1) is given by T = 0 : y+1 2 = x(1) 2x - y - 1 = 0 The equation of the family of circles touching the line 2x - y - 1 = 0 at (1,1) is: (x - 1)^2 + (y - 1)^2 + (2x - y - 1) = 0 Since the circle passes through the point (0,1) , substituting x = 0 and y = 1 gives: (0 - 1)^2 + (1 - 1)^2 + (2(0) - 1 - 1) = 0 1 - 2 = 0 = 1 2 Substituting = 1 2 into the equation of the circle: (x - 1)^2 + (y - 1)^2 + 1 2 (2x - y - 1) = 0 x^2 - 2x + 1 + y^2 - 2y + 1 + x -