A circle is inscribed between the parabola y = x^2 + 1 and the straight line x - y - 1 = 0 such that it touches both the curves and has the minimum possible area. The coordinates of the centre of this circle are

Step-by-step solution

The smallest circle touching both the parabola and the line will have its diameter along the common normal to both curves. First, we find the point on the parabola y = x^2 + 1 where the tangent is parallel to the line x - y - 1 = 0 . The slope of the given line is 1 . Differentiating the parabola's equation, we get dy dx = 2x . Setting the slope to 1 , we have 2x = 1 x = 1 2 . The corresponding y -coordinate is y = ( 1 2 )^2 + 1 = 5 4 . Thus, the point of contact on the parabola is A ( 1 2 , 5 4 ) . The normal to t