Question

Let C₁ and C₂ be the graphs of the function y = x² and y = 2x, 0 ≤ x ≤ 1 respectively. Let C₃ be the graph of a function y = f(x); 0 ≤ x ≤ 1, f(0) = 0. For a point P on C₁, let the lines through P parallel to the axis, meet C₂ and C₃ at Q and R, respectively (see Fig.). If for every position of P on (C₁), the areas of shaded region OPQ and ORP are equal, determine the function f(x).

Answer 1

See Fig.

On the curve C_1, that is, y = x² 

Let P be (α, α²). Hence, ordinate of point Q on C₂ is also α²

Now on C₂(y = 2x) the abscissa of Q is given by x = y/2 = α²/2.