Question

If [((4a²),(4a),(1)),((4b²),(4b),(1)),((4c²),(4c),(1))] [(f(-1)),(f(1)),(f(2))] = [(3a² + 3a),(3b² + 3b),(3c² + 3c)]

 and f(x) is a quadratic function and its maximum value occurs at a point V. A is a point of intersection of y = f(x) with x-axis and point B is such that the chord AB subtends a right angle at V. Find the area enclosed by f(x) and chord AB.

Answer 1

The given system of equations 

4x² f(–1) + 4x f(1) + f(2) = 3x² + 3x is satisfied for 3 distinct real numbers a, b and c

Comparing the co-efficients of x², x and constant terms, we get

4f(–1) = 3, 4f(1) = 3, f(2) = 0

Let f(x) = ax² + bx + c

Given f(–1) = 3/4 

a – b + c = 3/4

f(1) = 3/4 

a + b + c = 3/4

Thus, b = 0

And f(2) = 0

4a + 2b + c = 0

c = – 4a

Solving, we get

a = – (1/4) , b = 0, c = 1 

Thus, f(1) = – (1/4) x² + 1

Let the co-ordinates of B be B (t, l – (t²/4))

Let

t = – 8

Therefore, the co-ordinates of B = (– 8, –15).

Hence, the required area

= 125/3