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Conic Sections

Parabolas, ellipses, and hyperbolas, oh my! Learn about this eccentric bunch of shapes.

Level 5

         

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

Find \(|a+b+c+d+e+f|\).

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

Find \(a + b + c\).

A hyperbola \(H\) with equation \(xy=n\) (where \(n\le 1000\)) is rotated \(45^{\circ}\) to obtain the hyperbola \(H'\). Let the positive difference between the number of lattice points on \(H\) and \(H'\) be \(D\). Given that both \(H\) and \(H'\) have at least one lattice point, find the maximum possible value of \(D\).

Details and Asumptions: A lattice point is a point that has integer \(x\)- and \(y\)-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 \(a\) be the combined areas of the ellipse and the small circle, and let \(b\) be the area of the large circle.

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

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 \(8\) and \(27\). 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 \( \displaystyle \frac{a}{\sqrt{b}} \) where \( a\) and \(b\) are positive coprime integers and \(b\) is square-free .

Enter the value of \( a+b\).

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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