When Nine Points Are Worth But Eight: Euler’s Resolution of Cramer’s Paradox - Equations of Conics

Robert E. Bradley (Adelphi University) and Lee Stemkoski (Adelphi University)

In the 17th century, when analytic geometry was a brand new subject, mathematicians discovered that each of the conic sections may be expressed algebraically with an equation in the standard form \[\alpha y^2 + \beta xy + \gamma x^2 + \delta y + \varepsilon x + \zeta = 0.\]

For example, the equation of the circle \[(x - h)^2 + (y - k)^2 = r^2\] can be expanded and then expressed in the standard form above, with

\(\alpha = 1,\) \(\beta = 0,\) \(\gamma = 1,\)
\(\delta = -2k,\) \(\varepsilon = -2h,\) \(\zeta = h^2 + k^2 - r^2.\)

Similarly, the equation of the parabola \(y = ax^2 + bx + c\) can be expressed in standard form, where

\(\alpha = 0,\) \(\beta = 0,\) \(\gamma = a,\)
\(\delta = -1,\) \(\varepsilon = b,\) \(\zeta = c.\)

Furthermore, the type of conic section represented by an equation in standard form may be readily deduced from the coefficients of the equation. To do so, one first calculates the quantity \(\Delta=\beta^2 - 4 \alpha \gamma,\) known as the discriminant of the equation.

  • When \(\Delta > 0,\) the curve is a hyperbola.
  • When \(\Delta = 0,\) the curve is a parabola.
  • When \(\Delta < 0,\) the curve is an ellipse (and if also \(\alpha=\gamma\) and \(\beta = 0,\) then the ellipse is a circle).

In general, given 5 points on a conic section, we can find the corresponding equation of degree 2 in a way that is similar to the way we found the equation of a parabola in an earlier section. We substitute the values \((x_1, y_1),\) \((x_2, y_2),\) \((x_3, y_3),\) \((x_4, y_4),\) and \((x_5, y_5)\) into the equation \[\alpha y^2 + \beta xy + \gamma x^2 + \delta y + \varepsilon x + \zeta = 0.\] As before, we obtain a system of linear equations in the variables \(\alpha, \beta, \gamma, \delta, \varepsilon, \zeta.\) Then, provided that certain algebraic conditions regarding the points are satisfied, a unique solution can be found. As with the cases of the circle and the parabola, these algebraic conditions concern the collinearity of the points under consideration. These conditions were investigated by Euler, as we will see in the sections that follow.