Chapter 27 — Exercises

Work these by hand first; reach for Python only to check a number or draw a picture. The difficulty tiers are:

• ⭐ Foundational — read parameters off a standard equation; one definition, one step.
• ⭐⭐ Standard — combine two ideas (e.g., find $c$ from $a$ and $b$, then locate the foci).
• ⭐⭐⭐ Challenging — multi-step: build an equation from data, complete the square, or apply a property.
• ⭐⭐⭐⭐ Synthesis — connect conics to calculus, orbits, or another chapter's machinery.

A complete answer identifies the conic, gives every requested parameter, and shows the reasoning — not just a final number.

Part A — Identifying and Graphing Conics (⭐)

A1. ⭐ Identify each conic from its standard equation and state its center (or vertex). (a) $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$ (b) $\dfrac{x^2}{4} - \dfrac{y^2}{16} = 1$ (c) $x^2 = 12y$ (d) $x^2 + y^2 = 36$.

A2. ⭐ For the ellipse $\dfrac{x^2}{49} + \dfrac{y^2}{25} = 1$, state $a$, $b$, and whether the major axis is horizontal or vertical.

A3. ⭐ For the hyperbola $\dfrac{y^2}{9} - \dfrac{x^2}{16} = 1$, state $a$, $b$, the vertices, and which axis the branches open along.

A4. ⭐ Write the equation of the parabola with vertex at the origin and focus $(0, 2)$. Which direction does it open?

A5. ⭐ Sketch $\dfrac{x^2}{16} + \dfrac{y^2}{4} = 1$, marking the vertices on both axes and the foci. (Use §27.2 for the focus computation.)

A6. ⭐ True or false, with one sentence of justification: "In every ellipse the foci lie on the minor axis."

Part B — Foci, Directrix, and the $abc$ Relations (⭐⭐)

B1. ⭐⭐ For $\dfrac{x^2}{169} + \dfrac{y^2}{144} = 1$, find $c$, the foci, and the eccentricity $e$.

B2. ⭐⭐ For the hyperbola $\dfrac{x^2}{16} - \dfrac{y^2}{9} = 1$, find $c$, the foci, the eccentricity, and the equations of the asymptotes.

B3. ⭐⭐ A parabola is given by $y^2 = -20x$. Find $p$, the focus, the directrix, and the direction it opens.

B4. ⭐⭐ An ellipse has $\dfrac{x^2}{4} + \dfrac{y^2}{25} = 1$. Beware the trap of §27.2: identify $a$, $b$, $c$, the orientation of the major axis, and the foci.

B5. ⭐⭐ For $y = \dfrac{x^2}{8}$, rewrite in the form $x^2 = 4py$ and give the focus and directrix.

B6. ⭐⭐ A hyperbola has asymptotes $y = \pm \tfrac{3}{2}x$ and vertices $(\pm 4, 0)$. Find $a$, $b$, $c$, and write its standard equation.

B7. ⭐⭐ Compute the eccentricity of each and rank them from "roundest" to "most elongated": (i) $\dfrac{x^2}{100} + \dfrac{y^2}{36} = 1$, (ii) $\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1$, (iii) $x^2 + y^2 = 9$.

Part C — Building Conics from Data (⭐⭐⭐)

C1. ⭐⭐⭐ Find the standard equation of the ellipse with foci $(\pm 3, 0)$ and a vertex at $(5, 0)$.

C2. ⭐⭐⭐ Find the standard equation of the ellipse with foci $(0, \pm 4)$ that passes through $(3, 0)$.

C3. ⭐⭐⭐ Find the standard equation of the hyperbola with foci $(\pm 5, 0)$ and eccentricity $e = \tfrac{5}{3}$.

C4. ⭐⭐⭐ A parabola has focus $(0, -4)$ and directrix $y = 4$. Find its equation, vertex, and the direction it opens.

C5. ⭐⭐⭐ The latus rectum of a conic is the chord through a focus perpendicular to the major (or focal) axis. Show that for the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a>b$) the latus rectum has length $\dfrac{2b^2}{a}$. (Hint: set $x = c$ and solve for $y$.)

C6. ⭐⭐⭐ An ellipse has eccentricity $e = 0.6$ and semi-major axis $a = 10$. Find $c$, then $b$, and write its equation (major axis horizontal).

Part D — Translated and Rotated Conics (⭐⭐⭐)

D1. ⭐⭐⭐ Complete the square (the method of §27.7.1) to put $x^2 + 4y^2 - 6x + 8y + 9 = 0$ in standard form. Identify the conic, its center, and $a$ and $b$.

D2. ⭐⭐⭐ Complete the square: $9x^2 - 4y^2 - 36x - 8y + 68 = 0$. Identify the conic, its center, the vertices, and the asymptotes (as lines through the center).

D3. ⭐⭐⭐ Complete the square: $y^2 - 4x - 2y + 13 = 0$. Identify the conic, its vertex, focus, and directrix.

D4. ⭐⭐⭐ Use the discriminant $B^2-4AC$ (§27.7) to classify $4x^2 + 4xy + y^2 - 5x = 0$, then explain in one sentence what feature of the equation tells you it is rotated.

D5. ⭐⭐⭐⭐ Rotate the axes by $45°$ using $x = (u - v)/\sqrt2$, $y = (u + v)/\sqrt2$ to remove the cross term from $xy = 1$. Identify the resulting conic and its semi-axes. (This is a rectangular hyperbola in disguise.)

Part E — Reflective Properties and Tangents via Calculus (⭐⭐⭐⭐)

E1. ⭐⭐⭐ Using the "polarize" tangent formula from §27.8.1, find the tangent line to $\dfrac{x^2}{16} + \dfrac{y^2}{4} = 1$ at $(2\sqrt3, 1)$. (First verify the point is on the ellipse.)

E2. ⭐⭐⭐⭐ Find the tangent line to the hyperbola $\dfrac{x^2}{9} - \dfrac{y^2}{16} = 1$ at $(5, \tfrac{16}{3})$ by implicit differentiation (the technique of Chapter 8). Then confirm it matches the polarized form $\dfrac{x x_0}{9} - \dfrac{y y_0}{16} = 1$.

E3. ⭐⭐⭐⭐ A whispering gallery has an elliptical cross-section $\dfrac{x^2}{100} + \dfrac{y^2}{64} = 1$ (units in feet). Two people stand at the foci. How far apart are they, and what is the total path length a whisper travels from one to the other after a single wall reflection? (Use the focus-sum definition of §27.2.)

E4. ⭐⭐⭐⭐ For the parabola $x^2 = 8y$, a ray travels straight down (parallel to the axis) and strikes the curve at $(4, 2)$. The reflective property of §27.3 says it then passes through the focus. Verify by (a) locating the focus and (b) computing the tangent slope at $(4,2)$ via $\tfrac{dy}{dx}$, confirming the reflected ray indeed heads toward the focus.

E5. ⭐⭐⭐ Show that the tangent to the parabola $x^2 = 4py$ at a point $(x_0, y_0)$ on it can be written $x x_0 = 2p(y + y_0)$. (Differentiate implicitly, then simplify using $x_0^2 = 4p y_0$.)

Part F — Applied Conics: Orbits, Optics, Engineering (⭐⭐⭐⭐)

F1. ⭐⭐⭐ (Astronomy.) Mercury's orbit has semi-major axis $a = 0.387$ AU and eccentricity $e = 0.206$. Compute the perihelion distance $a(1-e)$ and aphelion distance $a(1+e)$ in AU.

F2. ⭐⭐⭐⭐ (Astronomy.) Halley's Comet has $a = 17.8$ AU, $e = 0.967$. Find its perihelion and aphelion. Neptune orbits at about $30$ AU; does Halley's aphelion exceed Neptune's orbit?

F3. ⭐⭐⭐ (Engineering — antennas.) A parabolic satellite dish has cross-section $x^2 = 4py$ and is $2$ m wide at the rim, where the rim sits $0.5$ m above the vertex. Find $p$ and the location of the focus (where the receiver goes). (The rim point $(1, 0.5)$ lies on the curve.)

F4. ⭐⭐⭐⭐ (Engineering — optics.) A solar concentrator is a paraboloid whose cross-section has focal length $p = 1.2$ m. A receiver tube is placed at the focus. Write the cross-sectional equation $x^2 = 4py$, and find how wide the dish is ($x$-span) at the height of the focus ($y = p$). (This width is the latus rectum.)

F5. ⭐⭐⭐⭐ (Acoustics.) A whispering gallery is an ellipse $30$ m long and $18$ m wide (full axes). Find the distance between the two foci, i.e., where the two listeners should stand.

F6. ⭐⭐⭐⭐ (Astrodynamics.) A spacecraft passes a planet on a path of eccentricity $e = 1.20$ relative to that planet (compare §27.4's 'Oumuamua discussion). Classify the path and state, in one sentence, what this eccentricity implies about whether the craft is captured or escapes.

Part G — Synthesis and Calculus Connections (⭐⭐⭐⭐)

G1. ⭐⭐⭐⭐ Using the area formula $A = \pi a b$ from §27.8.2, find the area enclosed by $4x^2 + 9y^2 = 36$. Then write (do not evaluate by hand) the arc-length integral for its perimeter using the parametrization $x = a\cos t$, $y = b\sin t$ (the arc-length machinery of Chapter 25), and name the type of integral it is (§27.8.3).

G2. ⭐⭐⭐⭐ A conic is given in the focus–directrix polar form of Chapter 26 by $r = \dfrac{8}{2 + 3\cos\theta}$. Rewrite it in the standard form $r = \dfrac{p}{1 + e\cos\theta}$, then identify the conic and its eccentricity. (Divide numerator and denominator by 2 first.)

G3. ⭐⭐⭐⭐ Earth's orbit has $a = 1$ AU, $e = 0.0167$. Write its focus-centered polar form $r = \dfrac{a(1 - e^2)}{1 + e\cos\theta}$ (Chapter 26), and use it to compute the perihelion ($\theta = 0$) and aphelion ($\theta = \pi$) distances. Confirm they match $a(1 \mp e)$.

G4. ⭐⭐⭐⭐ Derive the eccentricity from the asymptote slope: for a hyperbola with asymptote slope $m = b/a$, show $e = \sqrt{1 + m^2}$ (as in §27.4.3). Then find $e$ for the hyperbola whose asymptotes are $y = \pm 2x$.

G5. ⭐⭐⭐⭐ The full derivation that every inverse-square orbit is a conic requires solving a differential equation — the machinery of Chapter 19 — together with the focus-centered polar form of Chapter 26. Without solving it, explain in 3–4 sentences why eccentricity is the natural parameter that separates a bound planet (ellipse) from an escaping comet (hyperbola), citing $r = p/(1 + e\cos\theta)$ and what happens to $r$ as $\theta$ varies for $e < 1$ versus $e > 1$.

Tier Summary

Selected answers appear in appendices/answers-to-selected.md. Applied problems span astronomy, aerospace/antenna engineering, optics, acoustics, and astrodynamics.