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.

Let $${C_1}$$ and $${C_2}$$ be respectively, the parabolas $${x^2} = y - 1$$ and $${y^2} = x - 1$$. Let $$P$$ be any point on $${C_1}$$ and $$Q$$ be any point on $${C_2}$$. Let $${P_1}$$ and $${Q_1}$$ be the reflections of $$P$$ and $$Q$$, respectively, with respect to the line $$y=x$$. Prove that $${P_1}$$ lies on $${C_2}$$, $${Q_1}$$ lies on $${C_1}$$ and $$PQ \ge $$ min $$\left\{ {P{P_1},Q{Q_1}} \right\}$$. Hence or otherwise determine points $${P_0}$$ and $${Q_0}$$ on the parabolas $${C_1}$$ and $${C_2}$$ respectively such that $${P_0}{Q_0} \le PQ$$ for all pairs of points $$(P,Q)$$ with $$P$$ on $${C_1}$$ and $$Q$$ on $${C_2}$$.

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 $$AB$$.