Conic Sections: Circle, Ellipse, Parabola, Hyperbola
The four conics and how their standard forms reveal vertices, foci, and asymptotes at a glance.
The four conics and their signatures
- Circle: (x − h)² + (y − k)² = r² → center (h, k), radius r
- Ellipse: (x − h)²/a² + (y − k)²/b² = 1 → center (h, k); major along axis with LARGER denom
- Parabola: (x − h)² = 4p(y − k) (opens up/down) or (y − k)² = 4p(x − h) (left/right)
- Hyperbola: (x − h)²/a² − (y − k)²/b² = 1 (left-right) or (y − k)²/a² − (x − h)²/b² = 1 (up-down)
Quick-identify rule
- Same coefficients on x² and y² AND added → circle
- Different positive coefficients AND added → ellipse
- Squared on one variable, linear on the other → parabola
- Difference of two squares = 1 → hyperbola
Foci and eccentricity
| Conic | c relation | eccentricity e |
|---|---|---|
| Circle | c = 0 | e = 0 |
| Ellipse | c² = a² − b² | 0 < e < 1, e = c/a |
| Parabola | focus at distance p | e = 1 |
| Hyperbola | c² = a² + b² (note PLUS!) | e > 1, e = c/a |
⚠️ Sign trap
For ellipse, c² = a² − b². For hyperbola, c² = a² + b². Many students mix them up — memorize: ellipse subtracts, hyperbola adds.
Check yourself
📌 Identify
x²/9 − y²/16 = 1 is a:
Minus between squared terms → hyperbola. x² first → horizontal opening.
Practice with CBE-style practice questions
Pre-Calc Sem B practice for conics.