Geometry

# 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 ellipseā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 $$2y^2=2x-1$$ and $$2x^2=2y-1$$.

$$A,B,C,D$$ are consecutive vertices of a rectangle whose area is $$2006$$ square units. An ellipse with area $$2006\pi$$, which passes through $$A$$ and $$C$$ has its foci at $$B$$ and $$D$$.

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

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

Find the maximum area of $$\triangle POQ$$.

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

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