Let f be quadratic and a,b,c distinct real numbers satisfying the relation below for t=a,b,c. Let V be the maximum point of y=f(x). A is its positive x-intercept; B is on the curve, and angle AVB is a

Question

Let f be quadratic and a,b,c distinct real numbers satisfying the relation below for t=a,b,c. Let V be the maximum point of y=f(x). A is its positive x-intercept; B is on the curve, and angle AVB is a right angle. Find the area enclosed by chord AB and the curve. $4 t^{2} f (- 1) + 4 t f (1) + f (2) = 3 t^{2} + 3 t, t \in {a, b, c} .$

TestHub · Accepted Answer

The quadratic relation holds at three distinct values, so equating coefficients fixes f. Use positive intercept A and perpendicular slopes to find B. Integrate curve minus chord, then put u=x+8.

$f (- 1) = f (1) = 3/4, f (2) = 0, f (x) = 1 - x^{2} /4. V = (0, 1), A = (2, 0) . m_{V A} = - 1/2, m_{V B} = 2.$ $B = (h, 1 - h^{2} /4), - h /4 = 2, B = (- 8, - 15) . ℓ (x) = 3 x /2 - 3. f (x) - ℓ (x) = (2 - x) (x + 8) /4.$ $A = \int_{- 8}^{2} (f - ℓ) d x = \frac{1}{4} \int_{0}^{10} u (10 - u) d u = \frac{1}{4} [5 u^{2} - u^{3} /3]_{0}^{10} = 125/3.$

Mathematics - Area Under Curves Question with Solution | TestHub

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