Conic Sections
Chapter 11 — Precalculus
Conic sections — parabolas, ellipses, and hyperbolas — come from slicing a cone at different angles. This guide covers the standard forms, key features, and how to graph each one.
Chapter 11 Practice Problems
30+ questions on parabolas, ellipses, and hyperbolas
How to Identify Which Conic Section
Given a general equation Ax² + Cy² + Dx + Ey + F = 0 (no Bxy term):
Parabola
Only x² OR only y² (not both)
e.g.: y = x² + 2x − 3
Circle
x² and y² both present with same coefficient
e.g.: x² + y² = 25
Ellipse
x² and y² both present, same sign, different coefficients
e.g.: 4x² + 9y² = 36
Hyperbola
x² and y² both present with opposite signs
e.g.: x² − y² = 1
Completing the Square for Conics
Converting from general form to standard form requires completing the square. This reveals the center and key dimensions.
Example: Convert x² − 6x + y² + 4y = 3 to standard form
(x² − 6x + __) + (y² + 4y + __) = 3 + __ + __
(x² − 6x + 9) + (y² + 4y + 4) = 3 + 9 + 4
(x − 3)² + (y + 2)² = 16
Circle: center (3, −2), radius 4
The formula: (b/2)²
For x² + bx: add (b/2)² to both sides. For −6x: add (−6/2)² = 9. For +4y: add (4/2)² = 4.
Parabola
Standard Forms:
Opens up/down
(x − h)² = 4p(y − k)
Opens left/right
(y − k)² = 4p(x − h)
Key Parts:
Key tip: The focus and directrix are p units from the vertex. Every point on the parabola is equidistant from the focus and directrix — that's the definition.
Real-world applications: Satellite dishes, reflecting telescopes, car headlights, suspension bridges
Ellipse
Standard Forms:
Horizontal major axis
(x−h)²/a² + (y−k)²/b² = 1, a > b
Vertical major axis
(x−h)²/b² + (y−k)²/a² = 1, a > b
Key Parts:
Key tip: Remember: a > b always. The larger denominator goes under the variable whose axis is major. c² = a² − b² (subtract b², not add).
Real-world applications: Planetary orbits, whispering galleries, lithotripsy (kidney stones), optical lenses
Hyperbola
Standard Forms:
Opens left/right
(x−h)²/a² − (y−k)²/b² = 1
Opens up/down
(y−k)²/a² − (x−h)²/b² = 1
Key Parts:
Key tip: Key difference from ellipse: c² = a² + b² (add, not subtract). The hyperbola NEVER reaches its asymptotes but approaches them as x → ±∞.
Real-world applications: LORAN navigation, Cassegrain telescopes, cooling towers, sonic booms (intersection of shock waves)
Side-by-Side Comparison
| Feature | Parabola | Ellipse | Hyperbola |
|---|---|---|---|
| Sign pattern | Only x² or y² | + (same sign) | − (opposite) |
| c² formula | c² = a² (focus def.) | c² = a² − b² | c² = a² + b² |
| Number of foci | 1 | 2 | 2 |
| Asymptotes | None | None | y = ±(b/a)x |
| Eccentricity | e = 1 | 0 < e < 1 | e > 1 |
Practice Conic Sections Problems
Chapter 11 in NailTheTest has 30+ practice problems on parabolas, ellipses, and hyperbolas with step-by-step solutions and visual diagrams. Free to start.