Let $$S$$ be the focus of the parabola $${y^2} = 8x$$ and let $$PQ$$ be the common chord of the circle $${x^2} + {y^2} - 2x - 4y = 0$$ and the given parabola. The area of the triangle $$PQS$$ is

Consider the parabola $${y^2} = 8x$$. Let $${\Delta _1}$$ be the area of the triangle formed by the end points of its latus rectum and the point $$P\left( {{1 \over 2},2} \right)$$ on the parabola and $${\Delta _2}$$ be the area of the triangle formed by drawing tangents at $$P$$ and at the end points of the latus rectum. Then $${{{\Delta _1}} \over {{\Delta _2}}}$$ is

The line $$2x + y = 1$$ is tangent to the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$.

If this line passes through the point of intersection of the nearest directrix and the $$x$$-axis, then the eccentricity of the hyperbola is