Maybe that's too much work 3

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 $100000k^{2}$