Key Concepts of Conic Sections

Why This Matters

Conic sections aren't just abstract curves—they're the mathematical foundation for understanding everything from satellite orbits to the shape of a flashlight beam. You're being tested on your ability to recognize these curves from their equations, identify their key features (centers, foci, vertices, axes), and understand why each conic behaves the way it does. The unifying principle? Every conic section comes from slicing a cone at different angles, and that geometric origin determines everything about its algebraic properties.

Don't just memorize the four types and their equations. Focus on understanding eccentricity, the focus-directrix relationship, and how to convert between equation forms. These concepts let you classify any conic, graph it accurately, and solve application problems. When you see an equation on an exam, you should immediately know what shape you're dealing with and which features to find first.

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The Four Conic Types

Each conic section represents a different relationship between a plane and a double cone. The angle of intersection determines whether you get a closed curve, an open curve, or something in between.

Circle

• Eccentricity equals zero—the circle is the only conic with perfectly uniform curvature, making it the "baseline" for measuring how stretched other conics are
• Standard form is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius—the simplest conic equation to recognize
• All points equidistant from center—this defining property means circles have infinite lines of symmetry through the center

Ellipse

• Eccentricity between 0 and 1—the closer to 0, the more circular; the closer to 1, the more elongated the ellipse becomes
• Standard form is $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$, where $a$ is the semi-major axis and $b$ is the semi-minor axis—note both terms are positive and added
• Constant sum property—the sum of distances from any point on the ellipse to both foci always equals $2a$, which is how ellipses are physically constructed

Parabola

• Eccentricity equals exactly 1—this makes the parabola the boundary case between closed curves (ellipses) and open curves (hyperbolas)
• Standard forms are $y = a(x - h)^2 + k$ (vertical) or $x = a(y - k)^2 + h$ (horizontal)—only one squared term appears
• Focus-directrix definition—every point is equidistant from the focus and directrix, creating the reflective property used in satellite dishes and headlights

Hyperbola

• Eccentricity greater than 1—the larger the eccentricity, the more "open" the two branches become
• Standard form is $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$—the subtraction sign is your identifier; which term is positive determines horizontal vs. vertical opening
• Constant difference property—the absolute difference of distances from any point to both foci is constant at $2a$, contrasting with the ellipse's constant sum

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Classifying Conics from Equations

The general form and discriminant give you a systematic way to identify any conic without graphing it first.

General Form Equation

• General form is $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$—this is the "messy" form you'll often need to convert from
• Discriminant $B^2 - 4AC$ classifies the conic: negative for ellipse/circle, zero for parabola, positive for hyperbola
• Circle special case—when $A = C$ and $B = 0$, the general form represents a circle; otherwise a negative discriminant indicates an ellipse

Converting to Standard Form

• Complete the square—the essential technique for converting general form to standard form, grouping $x$ and $y$ terms separately
• Identify the center $(h, k)$ from the completed square form—this becomes your reference point for all other features
• Extract key parameters—once in standard form, read off $a$, $b$, and calculate $c$ using the appropriate relationship for that conic type

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Key Features and Parameters

Understanding these features lets you move from equation to graph and back—a skill tested repeatedly on exams.

Eccentricity

• Formula $e = \frac{c}{a}$ for ellipses and hyperbolas, where $c$ is the distance from center to focus—eccentricity quantifies "how non-circular" a conic is
• Classification values: circle ($e = 0$), ellipse ($0 < e < 1$), parabola ($e = 1$), hyperbola ($e > 1$)—memorize these boundaries
• Planetary orbits use eccentricity to describe orbital shape—Earth's orbit has $e \approx 0.017$, nearly circular

Focus and Directrix

• Focus is a fixed point that defines the conic's reflective properties—parabolas have one, ellipses and hyperbolas have two
• Directrix is a fixed line; for parabolas, every point is equidistant from focus and directrix—this is the defining property
• Relationship $c^2 = a^2 - b^2$ for ellipses and $c^2 = a^2 + b^2$ for hyperbolas—knowing which formula to use is critical

Center, Vertices, and Axes

• Center $(h, k)$ is the midpoint of the major axis for ellipses and the intersection of asymptotes for hyperbolas—circles and ellipses are centered; parabolas have no center
• Vertices are the endpoints of the major axis (ellipse) or the points where branches are closest (hyperbola)—located at distance $a$ from center along the major/transverse axis
• Axes of symmetry divide the conic into mirror-image halves—ellipses have two, parabolas have one, hyperbolas have two

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Graphing Strategies

Accurate graphing requires a systematic approach: identify the conic type, find key features, then sketch.

Graphing Process

• Start with center and vertices—plot $(h, k)$ first, then mark vertices at distance $a$ along the appropriate axis
• Locate foci using the $c$ value—for ellipses, foci lie between center and vertices; for hyperbolas, foci lie beyond vertices
• Draw asymptotes for hyperbolas—use the box method with dimensions $2a$ by $2b$ centered at $(h, k)$, then draw diagonals

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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 property differ?

2. Given the equation $4x^2 + 9y^2 - 16x + 18y - 11 = 0$, what type of conic is this, and how do you know before converting to standard form?

3. Compare and contrast how you would find the value of $c$ (distance from center to focus) for an ellipse versus a hyperbola with the same $a$ and $b$ values.

4. A conic has eccentricity $e = 0.8$. What type of conic is it, and what does this eccentricity value tell you about its shape compared to a circle?

5. If an FRQ asks you to write the equation of an ellipse given its foci at $(\pm 3, 0)$ and vertices at $(\pm 5, 0)$, what steps would you take and what is the final equation?