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 $$x = 0,{{dy} \over {dx}} = ..............$$

In a triangle $$ABC$$, $$a:b:c=4:5:6$$. The ratio of the radius of the circumcircle to that of the incircle is ............

A curve $$y=f(x)$$ passes through the point $$P(1, 1)$$. The normal to the curve at $$P$$ is $$a(y-1)+(x-1)=0$$. If the slope of the tangent at any point on the curve is proportional to the ordinate of the point, determine the equation of the curve. Also obtain the area bounded by the $$y$$-axis, the curve and the normal to the curve at $$P$$.