Geometry

Conic Sections

Conic Sections: Level 5 Challenges

         

A parabola whose vertex is the point V=(2,3)V=(2,3) and whose focus is (5,6)(5,6) has equation ax2+bxy+cy2+dx+ey+f=0ax^2+bxy+cy^2+dx+ey+f=0, where gcd(a,b,c,d,e,f)=1\gcd(a,b,c,d,e,f)=1.

Find a+b+c+d+e+f.\big|a+b+c+d+e+f\big|.

Given the ellipse 3x212x+2y2+12y+6=03x^{2} - 12x + 2y^{2} + 12y + 6 = 0, there exists a real number k=ab+ck = a\sqrt{b} + c, (where a,b,ca,b,c are all positive integers and bb is square-free), such that the hyperbola xy2y+3x=kxy - 2y + 3x = k is tangent to the ellipse at two points.

Find a+b+ca + b + c.

A hyperbola HH with equation xy=nxy=n (where n1000n\le 1000) is rotated 4545^{\circ} to obtain the hyperbola HH'. Let the positive difference between the number of lattice points on HH and HH' be DD. Given that both HH and HH' have at least one lattice point, find the maximum possible value of DD.

Details and Asumptions: A lattice point is a point that has integer xx- and yy-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 aa be the combined areas of the ellipse and the small circle, and let bb be the area of the large circle.

Find 10000ab\left\lfloor 10000\dfrac { a }{ b } \right\rfloor

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 88 and 2727. 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 ab \displaystyle \frac{a}{\sqrt{b}} where a a and bb are positive coprime integers and bb is square-free .

Enter the value of a+b a+b.

Details and Assumptions:

  • Latus rectum is the focal chord of a parabola which is perpendicular to the axis of the parabola.
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