If length of tangent at any point on the curve $$y = f(x)$$ intercepted between the point and the X-axis is of length 1. Find the equation of the curve.

Find the area bounded by the curves $$x^{2}=y, x^{2}=-y$$ and $$y^{2}=4 x-3$$.

If $$\left[\begin{array}{lll}4 a^{2} & 4 a & 1 \\ 4 b^{2} & 4 b & 1 \\ 4 c^{2} & 4 c & 1\end{array}\right]\left[\begin{array}{c}f(-1) \\ f(1) \\ f(2)\end{array}\right]=\left[\begin{array}{c}3 a^{2}+3 a \\ 3 b^{2}+3 b \\ 3 c^{2}+3 c\end{array}\right], \quad f(x)$$

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