Geometry

Conic Sections

Conic Sections: Level 3 Challenges

         

The figure above shows two ellipses whose major axes are perpendicular to each other. Each ellipse passes through the other ellipse’s foci, which form the vertices of a square. If the shaded square encloses an area of 16, then what is the area enclosed by one of the ellipses?

Find the shortest distance between the parabolas 2y2=2x12y^2=2x-1 and 2x2=2y12x^2=2y-1.

Note: Round off your answer to 2 decimal places.

A,B,C,DA,B,C,D are consecutive vertices of a rectangle whose area is 20062006 square units. An ellipse with area 2006π2006\pi, which passes through AA and CC has its foci at BB and DD.

If the perimeter of the rectangle can be expressed as pqp\sqrt{q} where pp is a positive integer and qq is a square-free positive integer, find p+qp+q.

Consider an ellipse x2144+y264=1\dfrac{x^2}{144} + \dfrac{y^2}{64} = 1. A line is drawn tangent to the ellipse at a point PP. A line segment drawn from the origin to a point QQ on this line is perpendicular to this tangent line.

Find the maximum area of POQ\triangle POQ.

The area of a circle centered at the origin, which is inscribed in the parabola y=x2100,y=x^2-100, can be expressed as abπ, \frac{a}{b} \pi, where aa and bb are coprime positive integers. What is the value of a+ba+b?

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