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.

A tangent to the ellipse x² + 4y² = 4 meets the ellipse x² + 2y² = 6 at P and Q. Prove that the tangents at P and Q of the ellipse x² + 2y² = 6 are at right angles.