Every conic, standard form by standard form
Circle, ellipse, hyperbola, and parabola — the equation that defines each one, and how to tell them apart.
Given a bare second-degree equation, classify it first with the A/C sign test before doing any algebra — it names which standard form you're headed toward. Then complete the square to match that standard form and read the conic's defining features straight off the constants.
Circles
No dedicated formula block — a circle is the ellipse you get when a = b.
Ellipses
Centered at the origin with a horizontal major axis, \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \qquad a > b > 0, with vertices (\pm a, 0), co-vertices (0, \pm b), and foci (\pm c, 0) where c^2 = a^2 - b^2. If the larger denominator sits under y^2 instead, the major axis is vertical and the foci lie on the y-axis.
Centered at (h, k), \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1.
The eccentricity is e = \dfrac{c}{a}, with 0 \le e < 1.
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Term
major axis
Term
minor axis
0 ≤ e < 1 for an ellipse — e = 0 is a circle.
Hyperbolas
Centered at the origin, transverse axis horizontal, \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, with vertices (\pm a, 0), foci (\pm c, 0) where c^2 = a^2 + b^2, and asymptotes y = \pm \dfrac{b}{a}x.
Transverse axis vertical, \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1, with vertices (0, \pm a), foci (0, \pm c), and asymptotes y = \pm \dfrac{a}{b}x.
Centered at (h, k), \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \quad\text{or}\quad \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1, with asymptotes passing through (h,k) instead of the origin.
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Because c^2 = a^2 + b^2 makes c always greater than a, every hyperbola's eccentricity exceeds 1. An ellipse's eccentricity, by contrast, always stays below 1.
Always e > 1, since c² = a² + b² forces c past a.
Term
asymptote
The hyperbola's box diagonals — the only conic here with any.
Parabolas
With vertex at the origin, x^2 = 4py \qquad \text{focus } (0, p), \ \text{directrix } y = -p, y^2 = 4px \qquad \text{focus } (p, 0), \ \text{directrix } x = -p.
With vertex at (h, k), (x-h)^2 = 4p(y-k) \qquad \text{focus } (h, k+p), \ \text{directrix } y = k-p, (y-k)^2 = 4p(x-h) \qquad \text{focus } (h+p, k), \ \text{directrix } x = h-p.
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Term
directrix
Classifying an unknown conic
Assume the second-degree equation has no xy term (B = 0): Ax^2 + Cy^2 + Dx + Ey + F = 0. Then, assuming the equation represents a genuine (non-degenerate) conic:
| condition on A and C | conic |
|---|---|
| A = C, same sign | circle |
| A \neq C, same sign | ellipse |
| A and C opposite signs | hyperbola |
| exactly one of A, C is zero | parabola |
Compares A and C in Ax² + Cy² + ... = 0 before you complete a single square.
Show proof
Each row falls straight out of the standard forms from the last two sections — the general equation is exactly what you get if you multiply a standard form out and shove every term to one side.
Circle. The standard form is (x-h)^2+(y-k)^2=r^2. Expand it and the coefficients of x^2 and y^2 are both exactly 1 (or whatever common factor the whole equation gets multiplied by) — identical numbers, same sign, by construction. A=C is not a coincidence; it's the algebraic fingerprint of "the two directions are treated identically," which is precisely what makes a circle round instead of oval.
Ellipse. The standard form \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 has different denominators whenever a \neq b. Multiplying through by a^2b^2 to clear denominators gives leading coefficients b^2 and a^2 — both positive (same sign), but unequal whenever a \neq b. An ellipse is a circle that got stretched unevenly, and the inequality A \neq C is the algebra recording that unevenness.
Hyperbola. The standard form \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 has a minus sign built into the definition itself — clearing denominators gives leading coefficients b^2 and -a^2, opposite signs no matter what a and b are. That minus sign isn't a computational accident; it's the entire geometric difference between a closed loop (ellipse) and two runaway branches (hyperbola).
Parabola. The standard forms y^2=4px and x^2=4py each contain only ONE squared variable. There's no term to write for the other one, which is the same as writing its coefficient as 0. A parabola isn't a squashed circle or a lopsided ellipse — it's a fundamentally different shape, with only one distance rule (focus and directrix) instead of two foci, and the missing squared term is exactly where that difference shows up in the algebra.