Chapter 27 — Quiz

Ten questions covering the three conics, their parameters, the unifying eccentricity, the reflective properties, and the calculus connections. Try each before opening the answer. Each answer cites the section to review.

1. A conic has eccentricity $e = 1$. Which curve is it, and what is the geometric condition that defines it?

AnswerA **parabola**. It is the set of points equidistant from a fixed focus and a fixed directrix line: $|PF| = \operatorname{dist}(P, \ell)$. This is the exact $e=1$ balance point of the focus–directrix family. *Section 27.3.*

2. For the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{25} = 1$, find $a$, $b$, $c$, the orientation of the major axis, and the foci.

AnswerThe larger denominator is $25$, under $y^2$, so the **major axis is vertical**: $a = 5$, $b = 3$. Then $c = \sqrt{a^2 - b^2} = \sqrt{25 - 9} = 4$. Foci lie on the major (vertical) axis at $(0, \pm 4)$ — a common spot to slip and place them on the $x$-axis. *Section 27.2.*

3. State the two $abc$ relations and the single sign that distinguishes them. Why does the sign make sense geometrically?

AnswerEllipse: $c^2 = a^2 - b^2$ (foci **inside**, $c < a$). Hyperbola: $c^2 = a^2 + b^2$ (foci **outside** the vertices, $c > a$). The plus sign for the hyperbola is forced because its foci sit *beyond* the vertices, so $c$ must be the largest length — only $a^2 + b^2$ makes $c$ exceed both $a$ and $b$. *Sections 27.2.2, 27.4.2.*

4. Find the focus and directrix of the parabola $x^2 = 12y$.

AnswerMatch to $x^2 = 4py$: $4p = 12$, so $p = 3$. The parabola opens upward with vertex $(0,0)$, **focus $(0, 3)$**, and **directrix $y = -3$**. *Section 27.3.2.*

5. For the hyperbola $\dfrac{x^2}{16} - \dfrac{y^2}{9} = 1$, give $c$, the eccentricity, and the asymptotes.

Answer$a = 4$, $b = 3$, so $c = \sqrt{a^2 + b^2} = \sqrt{16 + 9} = 5$. Eccentricity $e = c/a = 5/4 = 1.25$. Asymptotes $y = \pm \dfrac{b}{a}x = \pm \dfrac{3}{4}x$. *Section 27.4.*

6. The polar form $r = \dfrac{12}{1 + 2\cos\theta}$ describes which conic? Give its eccentricity and semi-latus rectum.

AnswerCompare to $r = \dfrac{p}{1 + e\cos\theta}$: here $e = 2$ and $p = 12$. Since $e > 1$, it is a **hyperbola**, with semi-latus rectum $p = 12$. This is the unified focus–directrix form from Chapter 26. *Section 27.5.*

7. A satellite dish, a car headlight, and a radio telescope all use the same conic. Which one, and what reflective property makes it the right shape?

AnswerThe **parabola**. Every ray parallel to the axis reflects through the focus (collecting: dish, telescope), and conversely every ray from the focus leaves parallel to the axis (projecting: headlight). Only the parabola focuses *all* parallel rays to a single point — a sphere suffers spherical aberration. *Section 27.3.4.*

8. Classify $5x^2 + 4xy + 5y^2 = 9$ using the discriminant, and explain what the $4xy$ term signals.

AnswerDiscriminant $B^2 - 4AC = 4^2 - 4(5)(5) = 16 - 100 = -84 < 0$, so it is an **ellipse**. The nonzero cross term $4xy$ signals that the ellipse is **rotated** (its axes are not aligned with the coordinate axes); a $45°$ rotation removes it. *Section 27.7.*

9. Earth's orbit has $a = 1$ AU and eccentricity $e = 0.0167$, with the Sun at one focus. Find the perihelion and aphelion distances, and state which conic the orbit is.

AnswerThe orbit is an **ellipse** (the Sun sits at a focus, by Kepler's first law). Perihelion $= a(1 - e) = 0.9833$ AU; aphelion $= a(1 + e) = 1.0167$ AU — a swing of about $3.4\%$ over the year. *Section 27.2.5.*

10. Using the polarized tangent rule of §27.8.1, write the tangent line to $\dfrac{x^2}{8} + \dfrac{y^2}{2} = 1$ at the point $(2, 1)$.

AnswerFirst check the point: $\tfrac{4}{8} + \tfrac{1}{2} = 1$. ✓ Replace $x^2 \to x x_0$ and $y^2 \to y y_0$: $\dfrac{x \cdot 2}{8} + \dfrac{y \cdot 1}{2} = 1$, i.e. $\dfrac{x}{4} + \dfrac{y}{2} = 1$, or $x + 2y = 4$ (slope $-1/2$). The polarize rule is just implicit differentiation (Chapter 8) packaged into a formula. *Section 27.8.1.*

Scoring Guide

  • 9–10 correct — Excellent. You can identify any conic, extract every parameter, and connect the geometry to calculus and orbits.
  • 7–8 correct — Solid. Re-check the $abc$ sign rule (Q3) and the major-axis orientation trap (Q2).
  • 5–6 correct — Review §27.2–27.5: the focus definitions, the two $abc$ relations, and the unifying polar form.
  • Below 5 — Re-read the chapter from §27.2, redo the "Check Your Understanding" boxes, then return to Part A and B of the exercises before retrying.