Let the parabola y = x^2 - kx + k^2 - 5 intersect the x-axis at two distinct points A and B. If the origin O (0,0) lies strictly inside the line segment AB and the point M (2,0) lies strictly outside the line segment AB, then the number of integral values of k is :

Step-by-step solution

Let f(x) = x^2 - kx + k^2 - 5 . Since the origin O (0,0) lies strictly inside the line segment AB, 0 lies between the roots of f(x) = 0 . For an upward opening parabola, this implies f(0) f(0) = k^2 - 5 Since 0 is between the roots, one root is negative and the other is positive. The point M (2,0) lies strictly outside the line segment AB, meaning 2 does not lie between the roots. As 2 > 0 , it must be greater than the positive root, which requires f(2) > 0 . f(2) = 4 - 2k + k^2 - 5 = k^2 - 2k - 1 > 0 The roots of