Let y = f(x) be a differentiable curve passing through the point M(1, -1). The chord joining any two points A(x1, f(x1)) and B(x2, f(x2)) on the curve always intersects the y-axis at N(0, 2*x1*x2). Which of the following is/are correct?

Question

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Integral from 0 to 1/2 of f(x) dx = 1/24. The chord AB meets the y-axis at (0, 2*x1*x2). Writing the line AB and evaluating at x=0 yields (g(x2)-g(x1))/(x2-x1) = -2 where g(x) = f(x)/x. So g is linear with slope -2: g(x) = -2x + c, hence f(x) = -2x² + cx. Using f(1) = -1: -2 + c = -1, so c = 1 and f(x) = x - 2x². The definite integral from 0 to 1/2 = [-2x³/3 + x²/2] from 0 to 1/2 = -1/12 + 1/8 = 1/24. Option B is correct. The area between the curve and x-axis (roots x=0 and x=1/2) equals 1/24, not 10/3 or 8/3, so options C and D are incorrect.

Let y = f(x) be a differentiable curve passing through the point M(1, -1). The chord joining any two points A(x1, f(x1)) and B(x2, f(x2)) on the curve always intersects the y-axis at N(0, 2x1x2). Which of the following is/are correct?

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