Questions: Conic Sections Overview

Question 1

A student sees these two equations: (x−2)²/9 + (y+1)²/4 = 1 and (x−2)²/9 − (y+1)²/4 = 1. Which correctly identifies each conic?

• A. Both are ellipses — both have the standard (x−h)²/a² form
• B. The first is an ellipse; the second is a hyperbola — addition of squares gives an ellipse, subtraction gives a hyperbola
• C. The first is a hyperbola; the second is an ellipse — the minus sign in the second means it opens outward
• D. Both are hyperbolas — any equation with two squared terms separated by a constant is a hyperbola

Answer: (B)

The sign between the two squared terms is the definitive test. Addition of two positive squared terms equals a positive constant → ellipse (or circle if denominators are equal). Subtraction → hyperbola. Option C reverses this. Options A and D reflect the common confusion of ignoring the sign between terms.

Question 2

A comet travels a path around the sun with eccentricity e = 1. What is the shape of its orbit?

• A. Ellipse — all bound orbits are ellipses
• B. Circle — e = 1 represents a perfect circle
• C. Parabola — e = 1 is the boundary between bound and unbound orbits
• D. Hyperbola — e > 0 means the orbit opens outward

Answer: (C)

Eccentricity unifies all conics: circle (e = 0), ellipse (0 < e < 1), parabola (e = 1), hyperbola (e > 1). A parabola is the knife-edge case between a closed orbit (ellipse) and an escape trajectory (hyperbola). The Sun sits at the focus of the parabola. Options A and B are wrong — circles require e = 0 and bound orbits require e < 1.

Question 3

Most conic section can be expressed as a function of x — that is, it passes the vertical line test.

• T. True
• F. False

Answer: False

A full ellipse or hyperbola fails the vertical line test: at any x value between the vertices, a vertical line intersects the curve at two points (one on the top half, one on the bottom). They are relations, not functions. Only a parabola opening left or right also fails as a function — and horizontal parabolas are conic sections too. Circles similarly fail. This is a key reason conics matter: they extend our study of curves beyond function graphs.

Question 4

The asymptotes of a hyperbola (x−h)²/a² − (y−k)²/b² = 1 pass through the center (h, k) of the hyperbola.

• T. True
• F. False

Answer: True

The asymptotes of this hyperbola are the lines y − k = ±(b/a)(x − h), which both pass through (h, k). They act as 'guides' that the branches of the hyperbola approach but never reach as x → ±∞. This is why the center is important even though no point of the hyperbola actually sits there — it's the intersection point of the asymptotes.

Question 5

What is the geometric unifying principle behind all four conic sections, and how does the concept of eccentricity capture it?

Model answer: All conics can be defined via a focus and directrix: a conic is the locus of points where the ratio of distance to focus to distance to directrix equals a constant e (the eccentricity). When e = 0, the curve closes into a circle; for 0 < e < 1, an ellipse; e = 1, a parabola; e > 1, a hyperbola. Alternatively, all four shapes arise from slicing a double cone at different angles — changing the tilt of the cutting plane transitions you between conics.

The focus-directrix definition reveals that circles, ellipses, parabolas, and hyperbolas are not four separate formulas but one geometric idea parameterized by eccentricity. This unification is essential for polar form of conics, which describes all four with a single compact equation r = ed/(1 − e cos θ). Understanding the underlying geometry is what allows you to recognize conics in physical applications — satellite dishes (paraboloids), planetary orbits (ellipses), and GPS signal geometry (hyperbolas).