Conic Sections Formulas

Why This Matters

Conic sections aren't just abstract curves—they're the mathematical foundation for understanding everything from planetary orbits to satellite dishes to the path of a thrown ball. In calculus and analytic geometry, you're being tested on your ability to recognize these curves from their equations, convert between forms, and apply their geometric properties to solve problems. The key concepts here are eccentricity, focus-directrix relationships, and standard vs. general form conversions.

Don't just memorize formulas. Every conic section is defined by a specific geometric relationship—a set of points satisfying a distance condition. When you understand why each equation looks the way it does, you can derive forgotten formulas on exam day and tackle FRQs that ask you to prove properties or find missing parameters. Know what concept each formula illustrates, and you'll be ready for anything.

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Closed Curves: Circles and Ellipses

These conics are defined by the sum of distances from points to one or two foci remaining constant. They form closed, bounded curves—the "gentler" members of the conic family.

Circle Equation

• Standard form: $(x - h)^2 + (y - k)^2 = r^2$—center at $(h, k)$ with radius $r$
• Defining property: all points are equidistant from the center, making the circle the simplest conic with eccentricity $e = 0$
• Special case: when $r = 0$, the equation represents a degenerate conic—just the single point $(h, k)$

Ellipse Equation

• Standard form: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$—centered at $(h, k)$ with semi-axes $a$ and $b$
• Defining property: the sum of distances from any point to the two foci equals $2a$ (the constant sum property)
• Orientation: if $a > b$, the major axis is horizontal; if $b > a$, it's vertical—know which denominator is larger

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Open Curves: Hyperbolas

Hyperbolas are defined by the difference of distances from points to two foci remaining constant. This creates two separate branches that extend infinitely—the "dramatic" conics.

Hyperbola (Horizontal Opening)

• Standard form: $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$—the positive term contains $x$, so it opens left and right
• Asymptotes: $y = k \pm \frac{b}{a}(x - h)$—these lines guide the curve's behavior at infinity
• Defining property: the absolute difference of distances to the foci equals $2a$ (the constant difference property)

Hyperbola (Vertical Opening)

• Standard form: $\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$—the positive term contains $y$, so it opens up and down
• Asymptotes: $y = k \pm \frac{a}{b}(x - h)$—notice the slope formula flips compared to horizontal hyperbolas
• Quick identification: whichever variable's term is positive determines the opening direction

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Single-Focus Curves: Parabolas

Parabolas are defined by equal distance from a point (focus) and a line (directrix). With eccentricity exactly $e = 1$, they sit at the boundary between ellipses and hyperbolas.

Parabola (Vertical Opening)

• Standard form: $(x - h)^2 = 4p(y - k)$—vertex at $(h, k)$, opens up if $p > 0$, down if $p < 0$
• Focus and directrix: focus at $(h, k + p)$, directrix at $y = k - p$—the vertex sits exactly halfway between them
• Parameter $p$: represents the focal length, the directed distance from vertex to focus

Parabola (Horizontal Opening)

• Standard form: $(y - k)^2 = 4p(x - h)$—vertex at $(h, k)$, opens right if $p > 0$, left if $p < 0$
• Focus and directrix: focus at $(h + p, k)$, directrix at $x = h - p$—same halfway relationship
• Key distinction: the squared variable determines orientation—squared $x$ = vertical; squared $y$ = horizontal

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Unifying Concepts: Eccentricity and General Form

These formulas connect all conic sections through shared geometric principles. Eccentricity measures how "stretched" a conic is, while the general form provides a universal equation for classification.

Eccentricity Formula

• Formula: $e = \frac{c}{a}$—where $c$ is the center-to-focus distance and $a$ is the center-to-vertex distance
• Classification by value: $e = 0$ (circle), $0 < e < 1$ (ellipse), $e = 1$ (parabola), $e > 1$ (hyperbola)
• Geometric meaning: eccentricity measures deviation from circularity—higher values mean more elongated or open curves

Focus-Directrix Definition

• Universal definition: $e = \frac{\text{distance to focus}}{\text{distance to directrix}}$—this ratio is constant for every point on the conic
• Applies to all conics: this single definition generates circles, ellipses, parabolas, and hyperbolas depending on $e$
• Exam application: use this when asked to derive equations or prove geometric properties from first principles

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Classification Tools: General Equation and Discriminant

When a conic isn't in standard form, these tools let you identify and convert it. The discriminant is your classification shortcut.

General Conic Section Equation

• Form: $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$—represents any conic section in a single equation
• Conversion strategy: use completing the square to convert to standard form (when $B = 0$) or rotation of axes (when $B \neq 0$)
• Coefficient roles: $A$, $B$, $C$ determine the conic type; $D$, $E$, $F$ affect position and size

Discriminant for Conic Classification

• Formula: $B^2 - 4AC$—calculate this value to instantly classify the conic type
• Classification rules: $B^2 - 4AC < 0$ → ellipse (or circle if $A = C$ and $B = 0$); $B^2 - 4AC = 0$ → parabola; $B^2 - 4AC > 0$ → hyperbola
• Exam strategy: this is your fastest tool for multiple-choice questions asking you to identify conic type from general form

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Quick Reference Table

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Self-Check Questions

1. Which two conic sections share the property of having two foci, and how does their defining distance relationship differ?

2. Given the equation $\frac{(x-2)^2}{9} + \frac{(y+1)^2}{16} = 1$, identify the conic type and determine whether the major axis is horizontal or vertical.

3. Compare and contrast the equations $(x-3)^2 = 8(y+2)$ and $(y+2)^2 = 8(x-3)$—what's the same, and what's different about these curves?

4. If you're given the general equation $2x^2 + 3y^2 - 4x + 6y - 1 = 0$, what's the fastest way to determine the conic type without completing the square?

5. A conic has eccentricity $e = 0.6$. What type of conic is it, and what does this eccentricity value tell you about its shape compared to a circle?