Let $$ABC$$ be an equilateral triangle inscribed in the circle $${x^2} + {y^2} = {a^2}$$. Suppose perpendiculars from $$A, B, C$$ to the major axis of the ellipse $$x.{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$, $$(a>b)$$ meets the ellipse respectively, at $$P, Q, R$$. so that $$P, Q, R$$ lie on the same side of the major axis as $$A, B, C$$ respectively. Prove that the normals to the ellipse drawn at the points $$P, Q$$ and $$R$$ are concurrent.

Consider the family of circles $${x^2} + {y^2} = {r^2},\,\,2 < r < 5$$. If in the first quadrant, the common taingent to a circle of this family and the ellipse $$4{x^2} + 25{y^2} = 100$$ meets the co-ordinate axes at $$A$$ and $$B$$, then find the equation of the locus of vthe mid-point of $$AB$$.

Find the co-ordinates of all the points $$P$$ on the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$, for which the area of the triangle $$PON$$ is maximum, where $$O$$ denotes the origin and $$N$$, the foot of the perpendicular from $$O$$ to the tangent at $$P$$.

The angle between a pair of tangents drawn from a point $$P$$ to the parabola $${y^2} = 4ax$$ is $${45^ \circ }$$. Show that the locus of the point $$P$$ is a hyperbola.