A parabola whose vertex is the point and whose focus is has equation , where .
Find
Given the ellipse , there exists a real number , (where are all positive integers and is square-free), such that the hyperbola is tangent to the ellipse at two points.
Find .
A hyperbola with equation (where ) is rotated to obtain the hyperbola . Let the positive difference between the number of lattice points on and be . Given that both and have at least one lattice point, find the maximum possible value of .
Details and Asumptions: A lattice point is a point that has integer - and -coordinates.
You may want to look at the list of Highly Composite Numbers.
A circle and an ellipse of the same area share the interior of a larger circle, without overlap.
For the size of the smaller circle, the ellipse has the largest possible area that could fit in the space between the smaller and larger circle. Let be the combined areas of the ellipse and the small circle, and let be the area of the large circle.
Find
You may want to use a computer for this.
Two mutually perpendicular chords are drawn from the vertex of parabola such that their lengths are and . This is possible for only one distance between the parabola's focus and the directrix.
The length of latus rectum of such a parabola can be expressed as where and are positive coprime integers and is square-free .
Enter the value of .
Details and Assumptions: