The set of all $$\alpha $$ $$ \in $$ R, for which w = $${{1 + \left( {1 - 8\alpha } \right)z} \over {1 - z}}$$ is purely imaginary number, for all z $$ \in $$ C satisfying |z| = 1 and Re z $$ \ne $$ 1, is :

Consider the following two binary relations on the set A = {a, b, c} :
R₁ = {(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)} and
R₂ = {(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)}.
Then :

If $$\overrightarrow a ,\,\,\overrightarrow b ,$$ and $$\overrightarrow C $$ are unit vectors such that $$\overrightarrow a + 2\overrightarrow b + 2\overrightarrow c = \overrightarrow 0 ,$$ then $$\left| {\overrightarrow a \times \overrightarrow c } \right|$$ is equal to :

If the tangents drawn to the hyperbola 4y² = x² + 1 intersect the co-ordinate axes at the distinct points A and B then the locus of the mid point of AB is :