Normals are drawn from the point $$P$$ with slopes $${m_1}$$, $${m_2}$$, $${m_3}$$ to the parabola $${y^2} = 4x$$. If locus of $$P$$ with $${m_1}$$ $${m_2}$$$$ = \alpha $$ is a part of the parabola itself then find $$\alpha $$.

Prove that, in an ellipse, the perpendicular from a focus upon any tangent and the line joining the centre of the ellipse to the point of contact meet on the corresponding directrix.

Let $$P$$ be a point on the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1,0 < b < a$$. Let the line parallel to $$y$$-axis passing through $$P$$ meet the circle $${x^2} + {y^2} = {a^2}$$ at the point $$Q$$ such that $$P$$ and $$Q$$ are on the same side of $$x$$-axis. For two positive real numbers $$r$$ and $$s$$, find the locus of the point $$R$$ on $$PQ$$ such that $$PR$$ : $$RQ = r: s$$ as $$P$$ varies over the ellipse.

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.