ABCD is a square of side length 2 units. $${C_1}$$ is the circle touching all the sides of the square ABCD and $${C_2}$$ is the circumcircle of square ABCD. L is a fixed line in the same plane and R is a fixed point.

If P is any point of $${C_1}$$ and Q is another point on $${C_2}$$, then

$${{P{A^2}\, + \,P{B^2}\, + P{C^2}\, + P{D^2}} \over {Q{A^2} + \,Q{B^2}\, + Q{C^2}\, + Q{D^2}}}$$ is equal to

The axis of a parabola is along the line $$y = x$$ and the distances of its vertex and focus from origin are $$\sqrt 2 $$ and $$2\sqrt 2 $$ respectively. If vertex and focus both lie in the first quadrant, then the equation of the parabola is

Let a hyperbola passes through the focus of the ellipse $${{{x^2}} \over {25}} + {{{y^2}} \over {16}} = 1$$. The transverse and conjugate axes of this hyperbola coincide with the major and minor axes of the given ellipse, also the produced of eccentricities of given ellipse and hyperbola is $$1$$, then

The equations of the common tangents to the parabola $$y = {x^2}$$ and $$y = - {\left( {x - 2} \right)^2}$$ is/are