Given Parabola, Find Axis

Professor McWorter brought to my attention the following problem: given a parabola, i.e., a curve in a plane which is a parabola. Use ruler and compass to find its axis.

The solution proceeds in a few steps. First, observe that only a ruler is needed to construct a tangent to a conic section at a point on that section.

Given two such tangents, they form a triangle with the base joining the points of tangency. The median to the base is shown to be parallel to the axis of the parabola. This actually a lemma due to Archimedes.

Having a line parallel to the axis, we draw a chord perpendicular to the line. The axis of the ellipse is the line perpendicular to the chord at its midpoint.

Professor McWorter credits Vitaly Bergelson of the Ohio State University with a simpler solution:

In some system of coordinates a parabola has the equation y = x². Imagine such coordinates in place. Draw two parallel lines meeting the curve in two places, with abscissas, say a and b for one line and c and d for the other. Then the slopes of the two lines are equal, which slopes are a + b and c + d. (This is from (b² - a²) / ( b - a) = a + b.) Hence the x-coordinates of the midpoints of the two chords the parallel lines form, namely (a + b)/2 and (c + d)/2, are also equal. Therefore, the line L joining these midpoints is parallel to the y-axis. Construct a line perpendicular to L and construct the midpoint of the corresponding chord. The line parallel to L through this midpoint is the y-axis and coincides with the axis of the parabola.

There is a shortcut that does not require algebra. In a circle, a line through the midpoints of two parallel chords passes through the center of the circle. By projective transformation, the same holds for ellipse and, in principle, for parabola as well. However, for parabola, the center is a point at infinity together with the second focus. It thus follows, that a line through the midpoints of two parallel chords is parallel to the axis of the parabola.

Conic Sections > Parabola

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