When you were in high school, you undoubtedly encountered conics at some point. The most familiar example is the parabola, introduced in algebra courses as the graph of the equation y = x2 or, more generally, y = ax2 + bx + c. You may have learned how to find the x– and y– intercepts, the coordinates of the vertex, and so on. These are all useful skills–although I personally have never used them outside of a classroom–but they tell you very little about the parabola as a geometrical object. The other conic sections–the ellipse and the hyperbola–don’t lend themselves to a description of the form y = f(x), hence are usually not studied at all. This is a terrible misfortune, as they are part of the treasure that comes down to us from the ancient geometers, and figure largely in physical sciences such as optics, acoustics, and celestial mechanics.
The most famous way to describe conics is as slices of cones, hence the name. In plane geometry, however, it is more convenient to think of them in a different way. We start with a line l (the directrix) and a point F not on the line (the focus). We select an eccentricity ε > 0. The conic is the locus of all points P such that the ratio of PF to the perpendicular distance from P to l is ε. If ε < 1, the conic is an ellipse; if ε = 1, a parabola; and if ε > 1, a hyperbola.
Here we see the eccentricity vary from 0 to ∞. The focus and directrix are in red. The parabola is the “boundary” between the family of hyperbolas and the family of ellipses, and occurs when the eccentricity is 1.
You may have observed that the ellipse becomes more and more “circular” as we slide the eccentricity toward 0. When the eccentricity reaches 0, then the circle collapses to the point F itself. But if we “blow up” the curve as the eccentricity shrinks to 0, then the limiting curve is indeed a circle. So we may view the circle as part of the same family of curves.
Many of the important properties of circles generalize to more universal properties shared by the other conics. For instance, from any point in the exterior of a conic there are precisely two tangents; or again, at each point on a conic there is exactly one tangent. It is possible to prove these properties by analytical methods, but this lacks geometrical finesse, involving arguments that would have been alien to the Greeks. The most excellent way to prove many of the basic theorems about conics is to use the eccentric circle.
Consider the conic with focus F, directrix l, and eccentricity ε. The interior of the conic is defined as the set of points P for which PF is less than ε times the distance to l. Let O be any point not in the interior and not on the directrix, and let s be its perpendicular distance from l. Construct the circle centered at O with radius ε⋅s.
Now let E be any point on the circle. Let R be the point where the line EF intersects l. Let P be the point where the line through F parallel to the radius intersects OR. An argument from similarity shows that P is on the conic.
Now imagine letting E traverse the circle. Then the conic is swept out by the point P as shown.
Here we have an ellipse. If the conic is a parabola, then the directrix is tangent to the eccentric circle; if the conic is a hyperbola, then the directrix intersects the eccentric circles at two points, and the two branches of the hyperbola correspond to the two parts of the circle.
The eccentric circle is what allows us to prove theorems concerning tangent lines. We already know that from any point in the exterior of a circle there are two tangents to the circle. Given an eccentric circle centered at O, the two tangents from F to the circle produce two tangents to the conic as shown.
Similarly, to show that there is exactly one tangent at a point of the conic, construct the eccentric circle at O. This circle passes through F, and there is exactly one tangent to the circle at F; this tangent produces the tangent at O.