Chapter 27 — Key Takeaways

The three conics at a glance

The ellipse fixes a sum and closes into a loop; the hyperbola fixes a difference and flies open into two branches; the parabola is the exact balance ($e=1$) between them. (§27.2–27.4)

The focus–directrix idea

Every conic can be defined by a single rule: a point $P$ such that the distance to a focus is $e$ times the distance to a directrix line. The number $e$ — the eccentricity — is which conic you have: $e<1$ ellipse, $e=1$ parabola, $e>1$ hyperbola. The parabola is the case where focus-distance exactly equals directrix-distance. (§27.3, §27.5)

The two $abc$ relations — the sign is everything

• Ellipse: $\;c^2 = a^2 - b^2\;$ — foci inside, $c < a$. ($a$ is the larger denominator.)
• Hyperbola: $\;c^2 = a^2 + b^2\;$ — foci outside the vertices, $c > a$.

In both, $e = c/a$. The plus sign for the hyperbola is forced because its foci lie beyond the vertices, so $c$ must be the largest length. Mixing these two up is the single most common error in the chapter. (§27.2.2, §27.4.2)

Eccentricity unifies all three

Chapter 26's focus-centered polar form makes the unity explicit:

$$r = \frac{p}{1 + e\cos\theta}, \qquad p = ed \;(\text{semi-latus rectum}).$$

One formula, every conic, controlled by the single dial $e$:

As $e$ rises continuously, the circle stretches into ellipses, the far focus escapes to infinity at $e=1$ to open a parabola, and beyond $e=1$ the curve splits into a hyperbola — no seams, one family. The same dial separates a bound orbit (ellipse) from an escaping one (hyperbola). (§27.5)

Reflective properties (and what each builds)

Only the parabola focuses all parallel rays to a single point; a sphere blurs them (spherical aberration). (§27.2.4, §27.3.4, §27.4.5)

Calculus connections

• Tangent by implicit differentiation (Chapter 8): for the ellipse, $\dfrac{2x}{a^2} + \dfrac{2y}{b^2}y' = 0 \Rightarrow y' = -\dfrac{b^2 x}{a^2 y}$. The clean result is the polarize rule — replace $x^2 \to x x_0$, $y^2 \to y y_0$: $\;\dfrac{x x_0}{a^2} + \dfrac{y y_0}{b^2} = 1$. (§27.8.1)
• Area of an ellipse via trig substitution (Chapter 16): $A = \pi a b$, collapsing to $\pi r^2$ when $a=b$. (§27.8.2)
• Perimeter via arc length (Chapter 25): $L = 4a\int_0^{\pi/2}\sqrt{1 - e^2\sin^2\theta}\,d\theta$, a non-elementary elliptic integral with no closed form — approximate numerically or with Ramanujan's formula. (§27.8.3)

Classification by discriminant

For $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, the discriminant $B^2 - 4AC$ classifies regardless of orientation:

• $B^2 - 4AC < 0$ → ellipse (circle if $A = C$, $B = 0$);
• $B^2 - 4AC = 0$ → parabola;
• $B^2 - 4AC > 0$ → hyperbola.

Linear terms $Dx + Ey$ signal a translation (remove by completing the square); a cross term $Bxy$ signals a rotation (remove by rotating the axes — the eigen-direction idea of Chapter 31). (§27.7)

Common pitfalls

• The $abc$ sign: ellipse $c^2 = a^2 - b^2$, hyperbola $c^2 = a^2 + b^2$. Don't swap them.
• $a$ is the larger ellipse denominator, always; the major axis (and the foci) lie along it. For $x^2/4 + y^2/25 = 1$ the foci are on the $y$-axis, not the $x$-axis.
• Parabola focal length is not the leading coefficient: $y = ax^2$ means $4p = 1/a$, so $p = 1/(4a)$ — not $p = a$.
• Hyperbola asymptote slopes are $\pm b/a$, not $\pm a/b$.

Skills you should now have

• Identify and sketch any conic from its standard or translated equation.
• Extract $a$, $b$, $c$, foci, vertices, directrix, asymptotes, and eccentricity.
• Build a conic's equation from focus/vertex/eccentricity data.
• Classify a general quadratic by discriminant; complete the square for translations.
• Find tangent lines by implicit differentiation and the polarize rule.
• Read an orbit's character (bound vs. escaping) off its eccentricity, and use the polar form.

Connections to the rest of the book

• Chapter 26 — the focus-centered polar form $r = p/(1+e\cos\theta)$ that unifies the conics.
• Chapter 8 — implicit differentiation, the engine behind the tangent formulas.
• Chapters 16–17 & 25 — trig substitution, arc length, and the non-elementary perimeter integral.
• Chapter 19 — the differential equation whose solution proves every inverse-square orbit is a conic.
• Chapters 30–31 — level-set gradients (the reflective-property proof) and the eigenvalue diagonalization of rotated quadratic forms.

What's next

Chapter 28 opens Part VI: Multivariable Calculus. Single-variable curves give way to vector-valued functions $\mathbf{r}(t)$ and surfaces $f(x,y)$. The conic family rises into three dimensions as the quadric surfaces — ellipsoids, paraboloids, hyperboloids — and the tools of this chapter (implicit tangents, eccentricity, diagonalized quadratic forms) generalize directly.

Reflection

Conic sections are the oldest topic in geometry still in daily working use. Apollonius catalogued them for beauty around 200 BC; Kepler and Newton found them written into the sky; today they shape telescopes, antennas, orbits, and medical devices. The deep idea is that they are not three curves but one family, turned by the single dial of eccentricity — and that same dial, in orbital mechanics, distinguishes a captured planet from an escaping comet. Geometry and algebra, as ever in this book, turn out to be one subject seen from two sides.