Summary: Conics Compared

This closing section gathers the chapter's results into two comparison tables and a short recap, so that the parallels and the (few but important) sign differences between the three conics are visible side by side.

The master comparison table places the circle, parabola, ellipse and hyperbola next to each other for: eccentricity, standard equation, tangent at a point $(x_1,y_1)$, the point of contact of a slope-$m$ tangent $y=mx+c$, and the condition of tangency itself. Reading down the "condition of tangency" column shows the pattern clearly: circle $c^2=a^2m^2+a^2=a^2(m^2+1)$; parabola $c=a/m$ (a special, simpler case since the parabola has no $b$); ellipse $c^2=a^2m^2+b^2$; hyperbola $c^2=a^2m^2-b^2$ — the ellipse and hyperbola conditions differ only in that one sign, which traces all the way back to the sign difference in their defining relations $b^2=a^2(1\mp e^2)$.

The auxiliary/director circle table shows that a parabola has neither an auxiliary nor a director circle (it has no centre for such a circle to be centred at), while the ellipse's director circle $x^2+y^2=a^2+b^2$ always exists, and the hyperbola's director circle $x^2+y^2=a^2-b^2$ exists only when $a>b$.

Let's Remember — the chapter's standing facts, restated:

• A conic is the locus of a point whose distance from a fixed focus is a constant ratio (the eccentricity $e$) of its distance from a fixed directrix.
• $e=1$: parabola. $0<e<1$: ellipse. $e>1$: hyperbola.
• The eccentricity of a rectangular hyperbola (one with $a=b$) is always $e=\sqrt2$. …