Find a point on the curve $${x^2} + 2{y^2} = 6$$ whose distance from
the line $$x+y=7$$, is minimum.

If $$P(1)=0$$ and $${{dp\left( x \right)} \over {dx}} > P\left( x \right)$$ for all $$x \ge 1$$ then prove that
$$P(x)>0$$ for all $$x>1$$.

If $${I_n}$$ is the area of $$n$$ sided regular polygon inscribed in a circle of unit radius and $${O_n}$$ be the area of the polygon circumscribing the given circle, prove that $$${I_n} = {{{O_n}} \over 2}\left( {1 + \sqrt {1 - {{\left( {{{2{I_n}} \over n}} \right)}^2}} } \right)$$$

Normals are drawn from the point $$P$$ with slopes $${m_1}$$, $${m_2}$$, $${m_3}$$ to the parabola $${y^2} = 4x$$. If locus of $$P$$ with $${m_1}$$ $${m_2}$$$$ = \alpha $$ is a part of the parabola itself then find $$\alpha $$.