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 need to recognize these shapes from their equations, convert between forms, and identify key features like centers, vertices, foci, and directrices. The equations 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; it's squared in the equation, so you'll often need to take a square root when solving
• Equal coefficients on both squared terms distinguish circles from ellipses when you see the equation in general form

Ellipse Equation

• Standard form: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$ — the sum of distances from any point on the ellipse to the two foci is constant
• The semi-major axis is always associated with the larger denominator, and the semi-minor axis with the smaller one. Whichever denominator is bigger tells you the orientation: if $a^2$ is under the $x$-term, the ellipse is wider horizontally; if it's under the $y$-term, it's taller vertically.
• Foci location uses $c^2 = a^2 - b^2$, where $a$ is the larger value. The foci sit along the major axis, each $c$ units from the 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 skill is identifying which term comes first, because that 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 pass through the center with slopes $\pm \frac{b}{a}$, forming 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 always comes first. Whichever variable's squared term is positive determines the opening direction.
• Foci location uses $c^2 = a^2 + b^2$ (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 $x = h - p$
• Recognition tip: whichever variable is squared determines the axis of symmetry. Squared $x$ = vertical axis of symmetry (opens up/down). Squared $y$ = horizontal axis of symmetry (opens left/right).

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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. Getting comfortable converting between forms is one of the most useful skills you can build for exams.

General Form of a Conic Section

General form: $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$

This is a universal equation that can represent any conic. When $B = 0$ (no $xy$ term, which is the typical case in Algebra 2), you classify by comparing $A$ and $C$:

• $A = C$ (same value, same sign) → Circle
• $A \neq C$ but same sign → Ellipse
• $A$ and $C$ have opposite signs → Hyperbola
• Either $A = 0$ or $C = 0$ (only one squared term) → Parabola

Converting to standard form requires completing the square. Group the $x$-terms together and the $y$-terms together, then complete the square for each group.

Eccentricity Formula

Formula: $e = \frac{c}{a}$ — this measures how "stretched" a conic is away from being circular.

• $e = 0$ → Circle (foci merge into the center)
• $0 < e < 1$ → Ellipse
• $e = 1$ → Parabola
• $e > 1$ → Hyperbola

Higher eccentricity means the foci are farther from the center relative to the vertices, producing a more elongated shape.

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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?