Maybe that's too much work 3

Calculus Level 5

Find the area of the region bounded by the following curves :

\[ y=f(x) , y=|g(x)| , x=0 \quad \text{and} \quad x=2 \]

f and g are two continuous functions satisfying the relations

  • \[ \displaystyle f(x+y) = f(x) + f(y) -8xy , \forall x,y \in R \]

  • \[ \displaystyle g(x+y) = g(x) + g(y) + 3xy , \forall x,y \in R \]

  • \[ \displaystyle f'(0)=8 , g'(0) = -4 \]

The answer is of the form \(\dfrac{a}{b}\) .

From any point K on the hyperbola \(\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}} = 1\) , three normals other than that at K are drawn . Now it is given that the centroid of the triangle formed by their feet lies on a hyperbola \[ \dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}} = k^{2} \] .

Evaluate the integer nearest to \(100000k^{2}\)

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