An ellipse has eccentricity $${1 \over 2}$$ and one focus at the point $$P\left( {{1 \over 2},1} \right)$$. Its one directrix is the common tangent, nearer to the point $$P$$, to the circle $${x^2} + {y^2} = 1$$ and the hyperbol;a $${x^2} - {y^2} = 1$$. The equation of the ellipse, in the standard form, is ............

Points $$A, B$$ and $$C$$ lie on the parabola $${y^2} = 4ax$$. The tangents to the parabola at $$A, B$$ and $$C$$, taken in pairs, intersect at points $$P, Q$$ and $$R$$. Determine the ratio of the areas of the triangles $$ABC$$ and $$PQR$$.

From a point $$A$$ common tangents are drawn to the circle $${x^2} + {y^2} = {a^2}/2$$ and parabola $${y^2} = 4ax$$. Find the area of the quadrilateral formed by the common tangents, the chord of contact of the circle and the chord of contact of the parabola.

If $$x{e^{xy}} = y + {\sin ^2}x,$$ then at $$x = 0,{{dy} \over {dx}} = ..............$$