Essential Conic Sections Equations

Why This Matters

Conic sections aren't just abstract curves—they're the mathematical foundation for everything from satellite dish design to planetary orbits. In Algebra 2, you're being tested on your ability to recognize these shapes from their equations, convert between forms, and identify key features like centers, vertices, foci, and directrices. The equations themselves encode geometric relationships: distance formulas, symmetry, and the interplay between variables.

What separates students who ace conic sections from those who struggle? Understanding why each equation looks the way it does. A circle uses addition because all points are equidistant from the center. A hyperbola uses subtraction because it describes the difference of distances. Don't just memorize formulas—know what geometric principle each equation represents and how changing parameters shifts, stretches, or flips the curve.

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Closed Curves: Circles and Ellipses

These conics share a key feature: they're bounded curves where the equation uses addition between the squared terms. The difference lies in whether the curve is perfectly round or stretched along one axis.

Circle Equation

• Standard form: $(x - h)^2 + (y - k)^2 = r^2$—all points are equidistant from center $(h, k)$
• Radius $r$ determines size; squaring it in the equation means you'll often need to take square roots when solving
• Equal coefficients on both squared terms distinguish circles from ellipses in general form

Ellipse Equation

• Standard form: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$—the sum of distances to two foci remains constant
• Semi-major axis $a$ is always the larger denominator; semi-minor axis $b$ is smaller (the bigger number determines orientation)
• Foci location uses $c^2 = a^2 - b^2$, where foci lie along the major axis at distance $c$ from center

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Open Curves: Hyperbolas

Hyperbolas are defined by the difference of distances to two foci, which is why their equations use subtraction. The key exam skill is identifying which term comes first—this determines whether the curve opens horizontally or vertically.

Hyperbola (Horizontal Transverse Axis)

• Standard form: $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$—branches open left and right
• Vertices lie at $(h \pm a, k)$, exactly $a$ units from the center along the x-axis
• Asymptotes have slopes $\pm \frac{b}{a}$, creating the boundary lines the branches approach but never touch

Hyperbola (Vertical Transverse Axis)

• Standard form: $\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$—branches open up and down
• The positive term comes first—whichever variable's squared term is positive determines the opening direction
• Foci location uses $c^2 = a^2 + b^2$ (note: addition, unlike ellipses), with foci along the transverse axis

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Single-Branch Curves: Parabolas

Parabolas have only one squared term, which is why they extend infinitely in one direction rather than closing. The parameter $p$ controls both the width of the curve and the location of the focus and directrix.

Parabola (Vertical Axis of Symmetry)

• Standard form: $(x - h)^2 = 4p(y - k)$—opens up when $p > 0$, down when $p < 0$
• Focus is at $(h, k + p)$, located inside the curve at distance $|p|$ from the vertex
• Directrix is the horizontal line $y = k - p$, always on the opposite side of the vertex from the focus

Parabola (Horizontal Axis of Symmetry)

• Standard form: $(y - k)^2 = 4p(x - h)$—opens right when $p > 0$, left when $p < 0$
• Focus is at $(h + p, k)$, and the directrix is the vertical line $x = h - p$
• Key recognition tip: whichever variable is squared determines the axis of symmetry (squared x = vertical axis; squared y = horizontal axis)

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Unifying Concepts: General Form and Classification

These tools help you identify and compare conics when they're not given in standard form. Mastering conversion between forms is a high-value exam skill.

General Form of a Conic Section

• Standard form: $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$—a universal equation that represents all conics
• Classification rule: if $B = 0$, compare $A$ and $C$: equal and same sign = circle; same sign but unequal = ellipse; opposite signs = hyperbola; one equals zero = parabola
• Converting to standard form requires completing the square—group x-terms and y-terms, then factor

Eccentricity Formula

• Formula: $e = \frac{c}{a}$—measures how "stretched" a conic is from circular
• Classification by eccentricity: $e = 0$ (circle), $0 < e < 1$ (ellipse), $e = 1$ (parabola), $e > 1$ (hyperbola)
• Geometric meaning: higher eccentricity means foci are farther from center relative to vertices, creating more elongated shapes

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Quick Reference Table

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Self-Check Questions

1. Given the equation $\frac{(x-2)^2}{16} + \frac{(y+1)^2}{16} = 1$, is this a circle or an ellipse? How do you know?

2. Compare the foci formulas for ellipses and hyperbolas. Why does one use subtraction and the other addition?

3. A parabola has vertex at $(3, -2)$ and focus at $(3, 1)$. Write its equation and identify whether it opens up, down, left, or right.

4. You're given $4x^2 - 9y^2 - 16x + 18y - 29 = 0$. Without completing the square, what type of conic is this? What feature of the equation tells you?

5. Compare and contrast: How do you determine the orientation of a hyperbola versus a parabola from their standard form equations?