Chapter 27 — Conic Sections: Ellipses, Parabolas, Hyperbolas

27.1 Two Thousand Years from Beauty to Necessity

Around 200 BC, Apollonius of Perga wrote eight books on a family of curves obtained by slicing a cone with a plane. The Greeks pursued these conic sections for their geometric beauty alone; the curves had no known use. Apollonius could not have imagined that he was cataloguing the shapes of planetary orbits, the cross-sections of every satellite dish, and the focusing geometry of every reflecting telescope.

Two thousand years later the payoff arrived. Kepler (1609) found that planets travel in ellipses. Galileo found that thrown projectiles travel in parabolas. Then Newton (1687) did something deeper than either: starting from the inverse-square law of gravity $F = -GMm/r^2$ and his own freshly invented calculus, he derived that every orbit under such a force is a conic section — ellipse, parabola, or hyperbola, with no other possibilities. The Greeks had, without knowing it, written down the complete list of trajectories the solar system is allowed to have.

This is one of the great vindications of the theme that runs through this whole book: geometry and algebra are inseparable, and both turn out to describe the physical world. A curve studied for its abstract elegance became, centuries later, the exact language of motion. This chapter tells that story while teaching you to read, write, and differentiate the three conics fluently.

Historical Note. Apollonius coined the three names from how a cutting plane meets the cone relative to its side: ellipse (Greek elleipsis, "falling short"), parabola (parabolē, "alongside / equal"), and hyperbola (hyperbolē, "exceeding / throwing beyond"). The same root gives us the rhetorical hyperbole — an exaggeration that "throws beyond" the truth. The vocabulary of geometry and the vocabulary of speech share an ancestor.

We will study conics through four lenses, each revealing something the others hide:

1. the focus–distance definition (sum, difference, or equality of distances);
2. the standard Cartesian equation ($x^2/a^2 \pm y^2/b^2 = 1$, $x^2 = 4py$);
3. the polar / focus–directrix form $r = p/(1+e\cos\theta)$ from Chapter 26, which unifies all three;
4. the original slice of a cone.

The Key Insight. A single number — the eccentricity $e$ — determines which conic you have and exactly how stretched it is. The Cartesian forms make the three look like different species, each with its own equation. Eccentricity reveals them as one continuous family: circle ($e=0$), ellipse ($01$). Hold that idea; everything in this chapter orbits it.

27.2 The Ellipse

27.2.1 The Focus–Sum Definition

Pick two fixed points $F_1$ and $F_2$ in the plane, the foci. An ellipse is the set of all points $P$ whose summed distance to the two foci is a fixed constant:

$$|PF_1| + |PF_2| = 2a.$$

The constant is written $2a$ because, as we will see, $a$ turns out to be the semi-major axis — the distance from the center to the far end of the longest diameter.

Geometric Intuition. This definition is literally a piece of string. Pin both ends of a loose string of length $2a$ at the two foci, pull a pencil against the string until it is taut, and slide the pencil all the way around. At every instant the two string segments are the distances $|PF_1|$ and $|PF_2|$, and they always sum to the fixed string length. Gardeners use exactly this trick — two stakes and a loop of rope — to lay out elliptical flower beds. The picture is the definition.

When the two foci coincide ($F_1=F_2$), the condition becomes "distance to one point is constant," which is a circle. A circle is an ellipse whose two foci have merged — the $e=0$ extreme of the family.

27.2.2 Deriving the Standard Cartesian Form

Watch the algebra turn the string definition into an equation. Place the center at the origin and the foci symmetrically on the $x$-axis at $F_1=(-c,0)$ and $F_2=(c,0)$. For a point $P=(x,y)$,

$$\sqrt{(x+c)^2+y^2} + \sqrt{(x-c)^2+y^2} = 2a.$$

Isolate one radical, square, simplify, isolate the remaining radical, and square again (a standard but tedious two-step). The cross terms cancel and you are left with

$$(a^2-c^2)x^2 + a^2 y^2 = a^2(a^2-c^2).$$

Because the longest a point can be from the center is $a$, and the foci sit inside at distance $cDefine $b^2 = a^2 - c^2$. Dividing through by $a^2 b^2$ gives the clean standard form:

$$\boxed{\ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\ }$$

with the geometric dictionary

• $a$ = semi-major axis (here along $x$, since $a>b$);
• $b$ = semi-minor axis (along $y$), where $b^2 = a^2 - c^2$;
• $c$ = center-to-focus distance, so $\boxed{c^2 = a^2 - b^2}$;
• $e = c/a$ = eccentricity, satisfying $0 \le e < 1$ for an ellipse.

The relation $c^2 = a^2 - b^2$ is the one to memorize with its picture: at the top of the ellipse, the point $(0,b)$ is equidistant from both foci, so each focal segment has length $a$ (their sum is $2a$). That segment, the semi-minor axis $b$, and the center-to-focus distance $c$ form a right triangle with hypotenuse $a$ — Pythagoras gives $a^2 = b^2 + c^2$ directly. The algebra and the triangle are the same fact.

Common Pitfall. Many students memorize "$c^2 = a^2 - b^2$" and then misapply it by assuming $a$ is whatever number sits under $x^2$. It is not — $a$ is always the larger of the two denominators, because the semi-major axis is by definition the longer one. For $x^2/4 + y^2/9 = 1$, the larger denominator is under $y^2$, so $a^2 = 9$, $b^2 = 4$, and the major axis is vertical. The foci sit on the major axis at $(0,\pm c)$ with $c=\sqrt{9-4}=\sqrt5$, not on the $x$-axis. Always identify the larger denominator first.

27.2.3 Worked Examples, Graduated

Example 1 (read off the parameters). For $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1$: the larger denominator is $9$, so $a=3$, $b=2$. Then $c = \sqrt{a^2-b^2} = \sqrt{9-4} = \sqrt5 \approx 2.236$, foci at $(\pm\sqrt5, 0)$, and $e = c/a = \sqrt5/3 \approx 0.745$. A moderately squashed ellipse.

Example 2 (a "rounder" ellipse). For $\dfrac{x^2}{100} + \dfrac{y^2}{64} = 1$: $a=10$, $b=8$, $c=\sqrt{100-64}=6$, $e=0.6$. Larger eccentricity than Example 1 would mean more elongated; here $e=0.6 < 0.745$, so this ellipse is rounder than Example 1's despite being physically bigger. Eccentricity measures shape, not size.

Example 3 (build the equation from data). Find the ellipse with foci $(\pm 4, 0)$ and a point $(0,3)$ on it. The foci give $c=4$ and a horizontal major axis. The point $(0,3)$ is the top of the minor axis, so $b=3$. Then $a^2 = b^2 + c^2 = 9 + 16 = 25$, so $a=5$ and the equation is $x^2/25 + y^2/9 = 1$, with $e = 4/5 = 0.8$.

Check Your Understanding. An ellipse has equation $\dfrac{x^2}{16} + \dfrac{y^2}{25} = 1$. Find $a$, $b$, $c$, the eccentricity, and the coordinates of the foci.

Answer

The larger denominator is $25$, so the major axis is vertical: $a=5$, $b=4$. Then $c = \sqrt{a^2-b^2} = \sqrt{25-16} = 3$. Eccentricity $e = c/a = 3/5 = 0.6$. Because the major axis is vertical, the foci lie on the $y$-axis at $(0, \pm 3)$ — a common spot to slip and put them on the $x$-axis.

27.2.4 The Reflective Property and Its Three Rigor Levels

Intuitive. A ray of light (or a sound wave) leaving one focus bounces off the ellipse and passes exactly through the other focus. Every ray, from every direction, reconverges. This is why an elliptical room becomes a whispering gallery: a whisper at one focus is inaudible across the room yet arrives crisply at the other focus, having been gathered by the whole wall. St. Paul's Cathedral in London, the Capitol's Statuary Hall in Washington, and the Mormon Tabernacle in Salt Lake City all exhibit this.

Computational. At a point $P$ on the ellipse, draw the two focal segments $PF_1$ and $PF_2$. The tangent line at $P$ makes equal angles with the two segments. Since the angle of incidence equals the angle of reflection, a ray arriving along $PF_1$ leaves along $PF_2$. You can verify the equal-angle claim coordinate by coordinate using the tangent slope we derive in §27.9.

Formal. Define $g(P) = |PF_1| + |PF_2|$. The ellipse is the level set $g=2a$. The gradient $\nabla g$ at $P$ is the sum of the two unit vectors pointing from the foci toward $P$; it is normal to the level set, hence normal to the tangent line. A sum of two unit vectors bisects the angle between them, so the normal bisects $\angle F_1 P F_2$, which forces the tangent to make equal angles with $PF_1$ and $PF_2$. The reflection law follows. This is the same level-set/gradient reasoning you will formalize for surfaces in Chapter 30.

Geometric Intuition. Why does a sum of distances create a focusing mirror? Because the ellipse is exactly the set of points where total path length $F_1 \to P \to F_2$ is constant ($=2a$). Fermat's principle says light travels paths of stationary length, and "constant length for every $P$" is the most stationary condition there is — every reflected ray takes the same time from focus to focus, so they all arrive in phase. The geometry and the optics are one statement.

Real-World Application — Lithotripsy (medicine). A kidney stone is shattered without surgery using an extracorporeal shock-wave lithotripter: the patient is positioned so the stone sits at one focus of a half-ellipsoid reflector, and a spark generates a shock wave at the other focus. The ellipse's reflective property concentrates the entire diffuse shock onto the stone, pulverizing it while sparing surrounding tissue. The same geometry that carries a whisper across a cathedral focuses enough energy to break rock inside a human body.

27.2.5 Application: Kepler's First Law

Kepler's First Law (1609): every planet moves in an ellipse with the Sun at one focus (not the center). Kepler extracted this from Tycho Brahe's decades of naked-eye observations of Mars; Newton later derived it from gravity. The full orbital-mechanics derivation — solving the inverse-square ODE — belongs to the differential-equations toolkit of Chapter 19, and the focus-centered polar form belongs to Chapter 26. Here we just read the geometry.

The eccentricity tells you the orbit's character:

Earth's orbit looks circular but is not: with the Sun at a focus, the closest approach (perihelion) is $a(1-e)$ and the farthest (aphelion) is $a(1+e)$, so the distance ratio is $(1+e)/(1-e) \approx 1.034$ — about a $3.4\%$ swing over the year. Halley's Comet, at $e=0.967$, dives inside Mercury's orbit at perihelion and retreats past Neptune at aphelion on a single 76-year ellipse.

Real-World Application — Satellite orbit design (aerospace). GPS satellites fly nearly circular orbits ($e\approx 0$) at altitude $\sim$20,200 km so their geometry stays predictable for timing. Molniya communications satellites do the opposite, deliberately choosing a high eccentricity ($e \approx 0.74$): by Kepler's second law a body moves slowly near aphelion, so a Molniya satellite "loiters" for hours over the high-latitude regions a geostationary satellite cannot reach. Engineers pick the eccentricity to buy the coverage they want.

27.3 The Parabola

27.3.1 The Focus–Directrix Definition

A parabola is the set of points $P$ equidistant from a fixed point (the focus $F$) and a fixed line (the directrix $\ell$):

$$|PF| = \operatorname{dist}(P,\ell).$$

This is the $e=1$ case of the unified focus–directrix definition from Chapter 26: the parabola is the exact balance point where distance-to-focus equals distance-to-directrix. Tip the balance toward the directrix ($e<1$) and you get an ellipse; tip it toward the focus ($e>1$) and you get a hyperbola.

27.3.2 Standard Cartesian Form

Put the vertex at the origin, the focus at $(0,p)$, and the directrix at $y=-p$. The equidistance condition for $P=(x,y)$ reads

$$\sqrt{x^2 + (y-p)^2} = y + p.$$

Square both sides: $x^2 + y^2 - 2py + p^2 = y^2 + 2py + p^2$. The $y^2$ and $p^2$ cancel, leaving

$$\boxed{\ x^2 = 4py\ }$$

which opens upward for $p>0$ and downward for $p<0$. Rotating the roles of $x$ and $y$ gives the sideways parabola $y^2 = 4px$ (opening right or left). The number $|p|$ is the focal length — the distance from vertex to focus.

Common Pitfall. When you see a parabola written as $y = ax^2$, the focal-length parameter is not $a$. Match it to the standard form: $y = ax^2$ means $x^2 = (1/a)\,y = 4py$, so $4p = 1/a$ and $p = 1/(4a)$. For $y = x^2$ this gives $p = 1/4$, so the focus sits at $(0, 1/4)$ — surprisingly close to the vertex. Students who guess "the focus is at $(0,1)$" or "$p=a$" go wrong here constantly.

27.3.3 Examples

• $y = x^2$ becomes $x^2 = y = 4py$, so $p = 1/4$; focus $(0, 1/4)$, directrix $y=-1/4$.
• $y = x^2/16$ becomes $x^2 = 16y = 4py$, so $p = 4$; focus $(0,4)$, directrix $y=-4$.
• $x^2 = -8y$ has $4p = -8$, so $p = -2$; opens downward, focus $(0,-2)$.

Check Your Understanding. A parabola opens to the right with vertex at the origin and focus at $(3,0)$. Write its equation and give its directrix.

Answer

A right-opening parabola has the form $y^2 = 4px$ with focus $(p,0)$. Here $p=3$, so $y^2 = 12x$. The directrix is the vertical line $x = -p = -3$. Sanity check: the vertex $(0,0)$ is distance $3$ from both the focus and the directrix, as it must be.

27.3.4 The Reflective Property

The parabola has the optical property that makes it the backbone of reflector design: every ray parallel to the axis reflects through the focus, and conversely every ray from the focus reflects out parallel to the axis. There is no second focus, because the directrix has, in effect, pushed the "other focus" off to infinity — incoming rays from an infinitely distant source arrive parallel.

This single property powers two opposite devices:

• Collecting. A satellite dish, radio telescope, or solar concentrator catches parallel rays from a far-off source and funnels them all onto a receiver placed at the focus. Crucially, only the parabola focuses all parallel rays to a single point. A spherical mirror focuses them to slightly different points — the blur called spherical aberration.
• Projecting. A car headlight or flashlight runs the same physics backward: put the bulb at the focus and the parabola sends the light out as a tight parallel beam.

Real-World Application — Radio astronomy and 5G (engineering). The 100-meter Green Bank Telescope is a parabolic dish that gathers radio waves from galaxies billions of light-years away onto a receiver a few centimeters across at its focus; the gain is enormous precisely because the paraboloid is the unique shape with no aberration. The same geometry, shrunk to a few centimeters, shapes the directional beams of 5G base stations and the parabolic feeds inside your microwave's radar-like proximity sensors.

Computational Note. The Hubble Space Telescope's primary mirror was ground to the wrong conic constant — off from the intended paraboloid by about $2.2$ micrometers at the edge. That tiny deviation was enough to scatter light away from the focus and blur every early image. The fix in 1993 (the COSTAR corrective optics) essentially inserted a compensating shape to restore the missing focusing. A few millionths of a meter on a conic section was a billion-dollar error.

27.4 The Hyperbola

27.4.1 The Focus–Difference Definition

A hyperbola is the set of points $P$ for which the difference of distances to two foci is a fixed constant:

$$\big|\,|PF_1| - |PF_2|\,\big| = 2a.$$

The outer absolute value is essential: it produces the hyperbola's two separate branches, one nearer each focus. Where the ellipse fixes a sum and closes up into a loop, the hyperbola fixes a difference and flies open to infinity.

27.4.2 Standard Cartesian Form

With foci at $(\pm c, 0)$, the difference condition reduces (by the same isolate-and-square procedure as the ellipse) to

$$\boxed{\ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\ }$$

with

• $a$ = center-to-vertex distance (vertices at $(\pm a, 0)$);
• $c^2 = a^2 + b^2$ — note the plus sign, the single most important difference from the ellipse;
• $e = c/a > 1$, since $c > a$;
• asymptotes $y = \pm \dfrac{b}{a}\,x$.

Common Pitfall. The sign in the $abc$ relation flips between the two closed-form conics, and mixing them up is the most frequent hyperbola error. For an ellipse, $c^2 = a^2 - b^2$ (foci inside, $chyperbola, $c^2 = a^2 + b^2$ (foci outside the vertices, $c>a$). Memory aid: in a hyperbola the foci sit beyond the vertices, so $c$ must be the largest length — and only $c^2 = a^2 + b^2$ makes $c$ bigger than both $a$ and $b$.

27.4.3 The Asymptotes and Why They Appear

Solve the standard form for $y$:

$$y = \pm b\sqrt{\frac{x^2}{a^2} - 1} = \pm \frac{b}{a}\,x\sqrt{1 - \frac{a^2}{x^2}}.$$

As $|x| \to \infty$, the factor $\sqrt{1 - a^2/x^2} \to 1$, so $y \to \pm(b/a)x$. The hyperbola hugs the two lines $y = \pm(b/a)x$ ever more closely without touching them. Those lines are the diagonals of the central rectangle of width $2a$ and height $2b$ — a sketching trick: draw that rectangle, draw its diagonals, and the hyperbola threads through the vertices toward the diagonals.

Geometric Intuition. The asymptotes are the hyperbola's "behavior at infinity," and they encode its eccentricity directly: $\tan(\text{asymptote angle from axis}) = b/a$, while $e = c/a = \sqrt{a^2+b^2}/a = \sqrt{1 + (b/a)^2}$. A nearly-flat asymptote ($b/a$ small) means $e$ just above $1$ and a sharply curved hyperbola; steep asymptotes mean large $e$ and a wide-open one. The two slopes you draw on the page are the eccentricity in disguise.

27.4.4 Examples

• $\dfrac{x^2}{9} - \dfrac{y^2}{16} = 1$: $a=3$, $b=4$, $c=\sqrt{9+16}=5$, $e=5/3\approx1.667$, asymptotes $y=\pm\frac43 x$.
• $x^2 - y^2 = 1$: $a=b=1$, $c=\sqrt2$, $e=\sqrt2\approx1.414$, asymptotes $y=\pm x$ — a rectangular (equiangular) hyperbola.

27.4.5 Reflective Property and Applications

A ray aimed at one focus of a hyperbola reflects off the near branch so that the outgoing ray appears to emanate from the other focus (its backward extension passes through it). This "virtual focus" behavior makes the hyperbola the standard secondary mirror in compound telescopes.

In a Cassegrain telescope, a large parabolic primary mirror concentrates starlight toward its focus; a small hyperbolic secondary intercepts the converging cone and redirects it cleanly through a hole in the primary to an eyepiece or sensor behind. The pairing works because the hyperbola shares one focus with the parabola and relays the light to its other focus — exactly the geometry the reflective property guarantees.

Real-World Application — Trajectories of unbound objects (astrodynamics). When a spacecraft passes a planet with more than escape energy, its path relative to that planet is a hyperbola with the planet at the focus: it sweeps in along one asymptote, whips around, and departs along the other, often stealing orbital energy in a gravity assist. Voyager 1 and 2 chained hyperbolic flybys of Jupiter and Saturn to fling themselves out of the solar system. And the interstellar visitor 1I/'Oumuamua (2017) was clocked at eccentricity $e \approx 1.20$ — unambiguously hyperbolic, the signature of an object that fell in from interstellar space, rounded the Sun once, and left forever.

27.5 The Unified View: Eccentricity and the Polar Form

Chapter 26 derived the focus–directrix definition in polar coordinates, with the focus at the origin:

$$r = \frac{p}{1 + e\cos\theta}, \qquad p = e\,d,$$

where $e$ is the eccentricity and $d$ is the distance from focus to directrix (so $p = ed$ is the semi-latus rectum, the half-width of the conic measured through the focus). One formula, every conic:

As $e$ climbs continuously from $0$, the curve morphs without any seam: the circle stretches into ellipses, the ellipses elongate until at $e=1$ the far focus escapes to infinity and the loop springs open into a parabola, and beyond $e=1$ the curve splits into the two branches of a hyperbola. There is no boundary where one "kind" of curve stops and another begins — only the single dial $e$ turning.

The Key Insight. This is the chapter's threshold idea made literal. The Cartesian forms ($x^2/a^2 + y^2/b^2 = 1$ versus $x^2 = 4py$ versus $x^2/a^2 - y^2/b^2 = 1$) look like three unrelated equation types and disguise the unity. The polar focus–directrix form $r = p/(1+e\cos\theta)$ exposes it: the conics are one object seen at different settings of a single parameter. This is also exactly the form orbital mechanics uses, with $e$ reading off whether an orbit is bound (ellipse), marginal (parabola), or escaping (hyperbola).

Check Your Understanding. A conic is given in polar form by $r = \dfrac{6}{1 + 2\cos\theta}$. Which conic is it, and what is its eccentricity?

Answer

Compare with $r = p/(1+e\cos\theta)$: here $e = 2$ and $p = 6$. Since $e > 1$, it is a hyperbola. (Bonus: the semi-latus rectum is $p = 6$, and $p = ed$ gives directrix distance $d = 3$.)

27.6 The Cone Itself

The name "conic section" records the original, pre-algebraic definition: each curve is the intersection of a plane with a double cone (two cones joined tip to tip, opening up and down). The tilt of the cutting plane, compared with the cone's own half-angle, selects the curve:

• plane perpendicular to the axis → circle;
• plane tilted less than the half-angle → ellipse (a closed slice through one nappe);
• plane parallel to a side of the cone → parabola (the borderline, where the slice just fails to close);
• plane tilted more than the half-angle → hyperbola (it cuts both nappes, giving two branches).

Notice the family structure reappears: tilt the plane continuously and the slice passes smoothly from circle through ellipses, hits the parabola at the exact parallel-to-side angle, then opens into hyperbolas. The eccentricity of §27.5 is, geometrically, a measure of this tilt.

Math Major Sidebar — From Greek geometry to Cartesian equations. Apollonius proved every such slice is one of these three curves with no calculus and no coordinates — only the "geometric algebra" of similar triangles and proportions on the cone. The bridge to the equations $x^2/a^2 \pm y^2/b^2 = 1$ waited nearly two millennia for Descartes (La Géométrie, 1637) to attach coordinates to the plane. The Dandelin spheres (1822) give a strikingly elegant modern proof that the planar slice satisfies the focus–distance definition: inscribe a sphere in the cone tangent to the cutting plane on each side; the two tangency points are precisely the foci, and the sphere–plane geometry forces the constant sum/difference of distances. It is one of the most beautiful proofs in elementary geometry, and it ties the cone definition to the focus definition with no algebra at all.

27.7 Algebraic Classification of the General Quadratic

Every conic, however translated or rotated, is the zero set of a general second-degree equation in two variables:

$$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0.$$

The discriminant $B^2 - 4AC$ classifies which conic it is, regardless of orientation:

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

(Degenerate cases — a single point, a pair of lines, or no real solutions — fill the seams.) The linear terms $Dx + Ey$ produce a translation of the center away from the origin, removed by completing the square. The cross term $Bxy$ signals a rotation of the axes, removed by rotating the coordinate frame.

27.7.1 Translated Conics (Completing the Square)

Example. Identify $4x^2 + 9y^2 - 16x + 18y - 11 = 0$. Group and complete the square in each variable:

$$4(x^2 - 4x) + 9(y^2 + 2y) = 11 \;\Rightarrow\; 4(x-2)^2 - 16 + 9(y+1)^2 - 9 = 11.$$

So $4(x-2)^2 + 9(y+1)^2 = 36$, i.e. $\dfrac{(x-2)^2}{9} + \dfrac{(y+1)^2}{4} = 1$ — an ellipse centered at $(2,-1)$ with $a=3$, $b=2$. Translation never changes the type of conic, only its location; the discriminant here is $0^2 - 4(4)(9) < 0$, confirming "ellipse" before we did any work.

27.7.2 Rotated Conics (Removing the Cross-Term)

Example. Classify and simplify $5x^2 + 4xy + 5y^2 = 9$.

The discriminant is $B^2 - 4AC = 16 - 100 = -84 < 0$, so it is an ellipse — tilted, because $B\neq 0$. Rotate the axes by $45°$ using $x = (u+v)/\sqrt2$, $y = (v-u)/\sqrt2$, and substitute:

$$5\cdot\frac{(u+v)^2}{2} + 4\cdot\frac{(u+v)(v-u)}{2} + 5\cdot\frac{(v-u)^2}{2} = 9.$$

Expanding, the $5(u+v)^2$ and $5(v-u)^2$ contribute $5u^2+5v^2$ each (the $\pm10uv$ cross terms cancel), while $4(u+v)(v-u) = 4(v^2-u^2)$. Collecting:

$$\frac{(5+5-4)u^2 + (5+5+4)v^2}{2} = \frac{6u^2 + 14v^2}{2} = 3u^2 + 7v^2 = 9,$$

so $\dfrac{u^2}{3} + \dfrac{v^2}{9/7} = 1$ — an ellipse with semi-axes $\sqrt3$ along the rotated $u$-axis and $3/\sqrt7$ along the $v$-axis. (A symbolic check of this expansion appears in §27.10.) The $45°$ rotation diagonalized the quadratic form; in Chapter 31 you will recognize this as finding the eigen-directions of the matrix $\begin{pmatrix}5&2\\2&5\end{pmatrix}$, whose eigenvalues $3$ and $7$ are exactly the coefficients that survive.

27.8 Conics and Calculus

The conics are not just static shapes — they are curves, and calculus is the study of curves. Two classic connections show the machinery of earlier chapters acting on them.

27.8.1 Tangent Lines by Implicit Differentiation

The standard conic equations define $y$ implicitly, so we differentiate implicitly (the technique of Chapter 8). For the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, differentiate both sides with respect to $x$:

$$\frac{2x}{a^2} + \frac{2y}{b^2}\,y' = 0 \;\Rightarrow\; y' = -\frac{b^2 x}{a^2 y}.$$

The slope is undefined at $y=0$ (the two ends of the major axis, where the tangent is vertical) and zero at $x=0$ (the top and bottom, where the tangent is horizontal) — exactly matching the picture. Plugging the slope at $(x_0,y_0)$ into point–slope form and simplifying using the fact that $(x_0,y_0)$ lies on the ellipse yields the remarkably clean tangent-line formula:

$$\boxed{\ \frac{x x_0}{a^2} + \frac{y y_0}{b^2} = 1\ }.$$

The rule is "polarize": replace $x^2$ by $x x_0$ and $y^2$ by $y y_0$ in the defining equation. The same trick gives the hyperbola's tangent, $\dfrac{x x_0}{a^2} - \dfrac{y y_0}{b^2} = 1$, and the parabola's. Implicit differentiation is the engine; the boxed formula is the payoff.

Check Your Understanding. Find the tangent line to the ellipse $\dfrac{x^2}{8} + \dfrac{y^2}{2} = 1$ at the point $(2, 1)$.

Answer

First confirm the point is on the curve: $4/8 + 1/2 = 1$. ✓ By the polarized formula, the tangent is $\dfrac{x\cdot 2}{8} + \dfrac{y\cdot 1}{2} = 1$, i.e. $\dfrac{x}{4} + \dfrac{y}{2} = 1$, or $x + 2y = 4$. (Check the slope against $y' = -\frac{b^2 x_0}{a^2 y_0} = -\frac{2\cdot2}{8\cdot1} = -\frac12$ — and indeed $x+2y=4$ has slope $-1/2$.)

27.8.2 Area of an Ellipse

The ellipse's area falls out of a single integral. The upper half is $y = b\sqrt{1 - x^2/a^2}$; double the area between $-a$ and $a$:

$$A = 2\int_{-a}^{a} b\sqrt{1 - \frac{x^2}{a^2}}\,dx = 4b\int_0^a \sqrt{1 - \frac{x^2}{a^2}}\,dx.$$

Use the trigonometric substitution $x = a\sin u$, $dx = a\cos u\,du$ (Chapter 16), so the radical becomes $\cos u$ and the limits run from $0$ to $\pi/2$:

$$A = 4ab\int_0^{\pi/2}\cos^2 u\,du = 4ab\cdot\frac{\pi}{4} = \boxed{\pi ab}.$$

When $a=b=r$ this collapses to the circle's $\pi r^2$, as it must. The ellipse is, in this sense, a circle scaled by $b/a$ in one direction, and area scales with the product of the two stretch factors.

27.8.3 The Perimeter: A Non-Elementary Integral

Area is easy; perimeter is famously hard. Parametrize the ellipse as $x = a\cos t$, $y = b\sin t$ and apply the arc-length formula (Chapter 25): with $\dot x = -a\sin t$, $\dot y = b\cos t$,

$$P = \int_0^{2\pi}\sqrt{a^2\sin^2 t + b^2\cos^2 t}\;dt = 4a\int_0^{\pi/2}\sqrt{1 - e^2\sin^2\theta}\;d\theta,$$

where $e$ is the eccentricity. This is the complete elliptic integral of the second kind, and here is the punchline: it has no antiderivative in elementary functions. No combination of polynomials, roots, exponentials, logarithms, and trig functions evaluates it. This is not a failure of cleverness — it is provably impossible, in the same spirit as the non-elementary integrals $\int e^{-x^2}dx$ we met in Chapter 14 and the techniques of Chapters 16–17.

So we approximate. Ramanujan's celebrated formula gives the perimeter to staggering accuracy:

$$P \approx \pi(a+b)\left[\,1 + \frac{3h}{10 + \sqrt{4 - 3h}}\,\right], \qquad h = \left(\frac{a-b}{a+b}\right)^2.$$

Historical Note. The stubborn impossibility of writing the ellipse's perimeter in closed form launched an entire branch of mathematics. Nineteenth-century giants — Legendre, then Abel and Jacobi — studied the inverse functions of these integrals, the elliptic functions, which generalize sine and cosine. What began as "I just want the circumference of an ellipse" grew into the theory that underlies modern algebraic geometry, the arithmetic of elliptic curves, and — by way of Wiles's elliptic-curve machinery — the 1995 proof of Fermat's Last Theorem. A perimeter you cannot integrate reshaped number theory.

Real-World Application — Orbit circumference and elliptic integrals (astrodynamics). To estimate how far a satellite travels in one revolution of an elliptical orbit, engineers must evaluate exactly this elliptic integral. There is no shortcut formula; mission-planning software computes it numerically (Gaussian quadrature) or via series. The same non-elementary integral that frustrated 18th-century geometers is a routine numerical call in every orbit-propagation library today — a clean illustration of the book's theme that hand computation builds understanding while machine computation builds power.

27.9 Computation: Visualizing and Verifying Conics

We follow the standard three-tier pattern: state the geometry, then let Python draw it and confirm the numbers. The first block plots all three conics side by side so you can see the family.

# Plot the three conic types and mark their foci, illustrating one family.
import numpy as np
import matplotlib.pyplot as plt

fig, axes = plt.subplots(1, 3, figsize=(15, 5))

# --- Ellipse: x²/9 + y²/4 = 1  (a=3, b=2, c=√5) ---
a, b = 3.0, 2.0
c = np.sqrt(a**2 - b**2)                 # c = 2.2360679...
t = np.linspace(0, 2*np.pi, 400)
axes[0].plot(a*np.cos(t), b*np.sin(t))
axes[0].plot([c, -c], [0, 0], 'ro')      # the two foci
axes[0].set_title(f'Ellipse  e = c/a = {c/a:.3f}')   # e = 0.745
axes[0].set_aspect('equal')

# --- Parabola: x² = 4py  with p = 1  (focus at (0,1)) ---
p = 1.0
x = np.linspace(-4, 4, 400)
axes[1].plot(x, x**2/(4*p))
axes[1].plot([0], [p], 'ro')             # focus
axes[1].axhline(-p, ls='--', color='gray')   # directrix y = -p
axes[1].set_title('Parabola  e = 1')
axes[1].set_aspect('equal')

# --- Hyperbola: x²/4 - y²/9 = 1  via cosh/sinh  (a=2, b=3, c=√13) ---
a2, b2 = 2.0, 3.0
c2 = np.sqrt(a2**2 + b2**2)               # note the PLUS sign: c² = a² + b²
s = np.linspace(-2, 2, 400)
axes[2].plot( a2*np.cosh(s), b2*np.sinh(s), 'b')   # right branch
axes[2].plot(-a2*np.cosh(s), b2*np.sinh(s), 'b')   # left branch
axes[2].plot([c2, -c2], [0, 0], 'ro')    # foci OUTSIDE the vertices
axes[2].set_title(f'Hyperbola  e = {c2/a2:.3f}')   # e = 1.803
axes[2].set_aspect('equal')

plt.tight_layout()
plt.show()
# Printed eccentricities: ellipse 0.745, parabola 1.000, hyperbola 1.803

The hyperbola is drawn with the parametrization $x = a\cosh s$, $y = b\sinh s$. It is a beautiful match: the identity $\cosh^2 s - \sinh^2 s = 1$ is exactly the equation $x^2/a^2 - y^2/b^2 = 1$ in disguise — the hyperbolic functions are to the hyperbola what sine and cosine are to the ellipse.

Computational Note. Notice the one-character difference that separates the two closed conics in code: the ellipse uses c = np.sqrt(a**2 - b**2) and the hyperbola uses c = np.sqrt(a**2 + b**2). The same sign that trips students on paper (the §27.4 pitfall) is the same sign that produces real foci inside an ellipse but outside a hyperbola. Get it wrong and the ellipse code throws a domain error the moment $b > a$ — the machine refuses to take the square root of a negative, which is itself a reminder that the foci of an ellipse must lie inside it.

We can also verify the rotated-conic computation of §27.7.2 symbolically with sympy, so we trust the hand algebra:

# Verify that the 45° rotation turns 5x² + 4xy + 5y² into 3u² + 7v².
import sympy as sp

u, v = sp.symbols('u v')
x = (u + v)/sp.sqrt(2)
y = (v - u)/sp.sqrt(2)
expr = 5*x**2 + 4*x*y + 5*y**2
print(sp.simplify(sp.expand(expr)))      # -> 3*u**2 + 7*v**2   ✓

Sympy confirms the cross-term vanishes and the surviving coefficients are $3$ and $7$, matching §27.7.2 exactly — hand and machine agree.

27.10 Applications: A Catalog

The conics are not a museum of antique curves; they are working tools across science and engineering. Grouped by field:

Astronomy and spaceflight. Planetary and lunar orbits (ellipses, Kepler). Comets: elliptical if gravitationally bound (Halley), hyperbolic if interstellar ('Oumuamua). Spacecraft escape trajectories (parabolic at exactly escape energy) and gravity-assist flybys (hyperbolic).

Optics and antennas. Parabolic reflectors in satellite dishes, radio telescopes, solar concentrators, headlights, and flashlights. Cassegrain telescopes (parabolic primary + hyperbolic secondary). Elliptical reflectors in some stage and studio lighting.

Acoustics and medicine. Elliptical whispering galleries (St. Paul's, the U.S. Capitol). Elliptical-reflector lithotripters that shatter kidney stones by focusing shock waves from one focus to the other.

Architecture and structures. Hyperbolic cooling towers (a hyperboloid of revolution — strong, material-efficient, and naturally drafting). Parabolic and catenary arches. (The St. Louis Gateway Arch is a catenary, close to but mathematically distinct from a parabola — a worthwhile reminder to check before labeling.)

Navigation. Hyperbolic positioning: LORAN and modern multilateration place a receiver on a hyperbola defined by the difference in signal arrival times from two transmitters — the focus-difference definition turned into a fix on a map. GPS solves an intersecting system of such surfaces.

A 2000-year-old catalog of curves still earns its keep in physics, engineering, medicine, and navigation.

27.11 Conics in the Modern Frontier

The reach extends into the most modern physics:

• Quantum mechanics. The hydrogen atom's electron feels a $1/r$ Coulomb potential — the same inverse-square form as gravity — so the classical orbits are conics, and that hidden symmetry explains the precise degeneracy pattern of the hydrogen spectrum.
• General relativity. Orbits in curved spacetime are almost ellipses but slowly precess; the tiny excess precession of Mercury's perihelion (43 arcseconds per century beyond Newton's prediction) was the first triumph of Einstein's theory in 1915.
• Particle physics. Relativistic charged particles in uniform magnetic fields trace conics in projection; bubble-chamber and detector tracks are read as conic arcs to recover momentum.

The thread from Apollonius to Einstein is unbroken: the curves were discovered for their geometry, and the universe keeps insisting on using them.

27.12 The Big Picture

Conic sections are a perfect small model of how mathematics works, and they reinforce the running themes of this book.

1. One object, many definitions. Focus-sum, focus-difference, focus–directrix, slice of a cone, second-degree equation — five descriptions of the same three curves, each exposing structure the others hide. Geometry and algebra are inseparable.

2. Equations encode geometry. The standard form $x^2/a^2 \pm y^2/b^2 = 1$ literally carries the foci, directrix, and eccentricity inside its coefficients. Reading geometry off algebra is Descartes's signature feat, and it is what made calculus on these curves possible.

3. A continuous family. Ellipse, parabola, and hyperbola are not three species but one curve at three settings of the eccentricity dial — and that same dial, in orbital mechanics, distinguishes a captured planet from an escaping comet.

4. Discovered for beauty, demanded by nature. Apollonius studied conics with no application in mind; Kepler and Newton found them written into the sky. Calculus appears in every quantitative field, and the conics were waiting in physics, optics, and medicine long before anyone went looking.

Add to Your Modeling Portfolio. Add a conic to your model and read its geometry through calculus. Track A — Biology: model the elliptical cross-section of a cell or vessel; compute its area $\pi ab$ exactly, then set up (and numerically evaluate) the elliptic-integral perimeter to compare with a circular approximation. Track B — Economics: a quadratic cost or utility function has conic level curves; classify one via the discriminant $B^2-4AC$ and find the tangent line (marginal trade-off) at a chosen point by implicit differentiation. Track C — Physics: propagate one orbit. Pick a body's $a$ and $e$, write the focus-centered polar form $r = a(1-e^2)/(1+e\cos\theta)$ (Chapter 26), and compute perihelion $a(1-e)$, aphelion $a(1+e)$, and the orbital path length via the elliptic integral. Track D — Data Science: generalize the conic to a quadratic form $\mathbf{x}^\top A\mathbf{x} = 1$ in $n$ dimensions. Its shape (ellipsoid vs. hyperboloid) is set by the eigenvalues of $A$ — exactly the $45°$-rotation idea of §27.7.2 scaled up. This is the geometry of a Gaussian's covariance ellipse and of ridge-regression level sets.

Looking Ahead

Chapter 28 opens Part VI: Multivariable Calculus. We leave single-variable curves behind and work with vector-valued functions $\mathbf{r}(t)$ tracing paths in space, then functions $f(x,y)$ graphed as surfaces. Everything you have built generalizes: derivatives become gradients (Chapter 30), the implicit differentiation of §27.8 becomes the level-set normal vector, the diagonalization of the rotated conic in §27.7.2 becomes the eigenvalue analysis of quadratic forms (Chapter 31), and the conics themselves grow into quadric surfaces — ellipsoids, paraboloids, hyperboloids — in three dimensions. The dimension rises; the calculus framework adapts, carrying the conic family upward with it.

Reflection

For two thousand years the conic sections were an exercise in pure geometry, beautiful and useless. Then Kepler looked at Mars, Newton looked at gravity, and the curves snapped into place as the exact shapes of motion. A whisper crosses a cathedral, a shock wave shatters a kidney stone, a dish gathers a galaxy's faint radio hiss, a spacecraft slings itself out of the solar system — all of it is Apollonius's geometry, read through Newton's calculus. You can now identify any conic from its equation, extract its foci and eccentricity, find its tangent lines by implicit differentiation, and recognize why its perimeter resists every elementary formula. Most of all, you can see the unity: not three curves, but one family, turning under a single parameter. Turn the page; in the next chapter the plane becomes space.