Prove that
$${2^k}\left( {\matrix{ n \cr 0 \cr } } \right)\left( {\matrix{ n \cr k \cr } } \right) - {2^{^{k - 1}\left( {\matrix{ n \cr 2 \cr } } \right)}}\left( {\matrix{ n \cr 1 \cr } } \right)\left( {\matrix{ {n - 1} \cr {k - 1} \cr } } \right)$$
$$ + {2^{k - 2}}\left( {\matrix{ {n - 2} \cr {k - 2} \cr } } \right) - .....{\left( { - 1} \right)^k}\left( {\matrix{ n \cr k \cr } } \right)\left( {\matrix{ {n - k} \cr 0 \cr } } \right) = {\left( {\matrix{ n \cr k \cr } } \right)^ \cdot }$$

If a, b, c are in A.P., $${a^2}$$, $${b^2}$$, $${c^2}$$ are in H.P., then prove that either a = b = c or a, b, $${ - {c \over 2}}$$ form a G.P.

For the circle $${x^2}\, + \,{y^2} = {r^2}$$, find the value of r for which the area enclosed by the tangents drawn from the point P (6, 8) to the circle and the chord of contact is maximum.

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 $$\alpha $$.