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How did conic sections help ancient Greek mathematicians address the problem of doubling the cube?

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Diagram showing a double cone intersected by planes at different angles, producing the conic sections: a horizontal plane creates a circle, a tilted plane creates an ellipse, a plane parallel to the cone's side forms a parabola, and a steep plane produces a hyperbola. Each section is labeled, with the axis and apex of the cone indicated. — analytic geometry The conic sections result from intersecting a plane with a double cone, as shown in the figure. There are three distinct families of conic sections: the ellipse (including the circle), the parabola (with one branch), and the hyperbola (with two branches).

The ancient Greek mathematician Menaechmus is credited with discovering conic sections and applying them to solve the problem of doubling the cube, which was impossible to solve using only a straight edge and compass.123

The problem required constructing a cube with twice the volume of a given cube, which Hippocrates of Chios demonstrated was equivalent to finding two mean proportionals between a line segment $a$ and its double $2 a$ . Mathematically, this requires finding $x$ and $y$ such that:
$a : x = x : y = y : 2 a$

This proportion is equivalent to solving the following equations simultaneously:
* $x^{2} = a y$
* $y^{2} = 2 a x$
* $x y = 2 a^{2}$

These equations represent two parabolas and a hyperbola, respectively. By finding the intersection points of these conic sections, the Greeks were able to geometrically determine the values of $x$ and $y$ necessary to construct the required cube.32