The locus of the mid-point of the line segment joining the focus of the parabola y² = 4ax to a moving point of the parabola, is another parabola whose directrix is :

The system of linear equations
3x - 2y - kz = 10
2x - 4y - 2z = 6
x+2y - z = 5m
is inconsistent if :

Let P = $$\left[ {\matrix{ 3 & { - 1} & { - 2} \cr 2 & 0 & \alpha \cr 3 & { - 5} & 0 \cr } } \right]$$, where $$\alpha $$ $$ \in $$ R. Suppose Q = [ q_ij] is a matrix satisfying PQ = kl₃ for some non-zero k $$ \in $$ R.
If q₂₃ = $$ - {k \over 8}$$ and |Q| = $${{{k^2}} \over 2}$$, then a² + k² is equal to ______.

The minimum value of $$\alpha $$ for which the
equation $${4 \over {\sin x}} + {1 \over {1 - \sin x}} = \alpha $$ has at least one solution in $$\left( {0,{\pi \over 2}} \right)$$ is .......