An ellipse intersects the hyperbola $$2{x^2} - 2{y^2} = 1$$ orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinate axes then

In a triangle $$ABC$$ with fixed base $$BC$$, the vertex $$A$$ moves such that $$$\cos \,B + \cos \,C = 4{\sin ^2}{A \over 2}.$$$

If $$a, b$$ and $$c$$ denote the lengths of the sides of the triangle opposite to the angles $$A, B$$ and $$C$$, respectively, then

Let $$P\left( {{x_1},{y_1}} \right)$$ and $$Q\left( {{x_2},{y_2}} \right),{y_1} < 0,{y_2} < 0,$$ be the end points of the latus rectum of the ellipse $${x^2} + 4{y^2} = 4.$$ The equations of parabolas with latus rectum $$PQ$$ are :

On the ellipse $$4{x^2} + 9{y^2} = 1,$$ the points at which the tangents are parallel to the line $$8x = 9y$$ are