Conic Section Formulas

Why This Matters

Conic sections are the mathematical foundation for everything from satellite orbits to headlight reflectors to arches in architecture. In College Algebra, you need to recognize these shapes from their equations, identify key features like centers, vertices, and foci, and understand how the parameters control the shape.

All four conic sections come from the same geometric idea (slicing a cone), so their equations share a logical structure. The real exam skill is knowing what makes each conic unique: circles have equal distances in all directions, ellipses stretch that circle, parabolas have a single focus, and hyperbolas split into two branches. When you understand why each equation looks the way it does, you can reconstruct forgotten formulas and tackle graph-to-equation problems with confidence.

---

The Foundation: Circles and the Distance Principle

Every conic section relates to distance in some way. The circle is the simplest case: every point is exactly the same distance from a fixed center. This equal-distance principle is your conceptual anchor for understanding how the other conics modify it.

Circle

• Standard form: $\boldsymbol{(x - h)^2 + (y - k)^2 = r^2}$ represents all points equidistant from center $(h, k)$
• Radius $r$ determines size. Because it's squared in the equation, you'll often need to take a square root when solving. For example, if $r^2 = 49$, then $r = 7$.
• Center coordinates $(h, k)$ shift the circle from the origin. Watch the signs carefully: $(x - 3)^2$ means $h = 3$, and $(x + 2)^2$ means $h = -2$ because $(x + 2) = (x - (-2))$.

---

Stretched Circles: Ellipses

An ellipse is what happens when you stretch a circle unevenly. The sum of distances from any point on the ellipse to two foci is constant. The key to ellipse problems is identifying which axis is longer, because that tells you whether the ellipse is horizontal or vertical.

Ellipse (Horizontal Major Axis)

• Standard form: $\boldsymbol{\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1}$ where $a > b$. The larger denominator sits under the $x$-term.
• Semi-major axis $a$ stretches horizontally; semi-minor axis $b$ compresses vertically.
• Foci lie along the major axis at distance $c$ from the center, where $c^2 = a^2 - b^2$.

Ellipse (Vertical Major Axis)

• Standard form: $\boldsymbol{\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1}$ where $a > b$. The larger denominator sits under the $y$-term.
• Semi-major axis $a$ stretches vertically; the ellipse is taller than it is wide.
• Same foci relationship applies: $c^2 = a^2 - b^2$, and the foci lie along the vertical axis.

---

Single-Focus Curves: Parabolas

Parabolas are unique among conics because they have exactly one focus and extend infinitely in one direction. Every point on a parabola is equidistant from the focus and a line called the directrix. The parameter $p$ controls both how "wide" the parabola opens and where the focus sits.

Parabola (Vertical Opening)

• Standard form: $\boldsymbol{(x - h)^2 = 4p(y - k)}$. Opens upward if $p > 0$, downward if $p < 0$.
• Vertex at $(h, k)$ is the turning point; focus sits at $(h, k + p)$.
• Directrix is the horizontal line $y = k - p$, located $|p|$ units on the opposite side of the vertex from the focus.

Parabola (Horizontal Opening)

• Standard form: $\boldsymbol{(y - k)^2 = 4p(x - h)}$. Opens right if $p > 0$, left if $p < 0$.
• Vertex at $(h, k)$ remains the turning point; focus sits at $(h + p, k)$.
• Directrix is the vertical line $x = h - p$. It's always on the opposite side of the vertex from the focus.

---

Two-Branch Curves: Hyperbolas

Hyperbolas look like two parabolas facing away from each other, but they follow a different distance rule. The absolute difference of distances from any point on the hyperbola to two foci is constant. The subtraction sign between the two fraction terms is your instant identifier for hyperbolas.

Hyperbola (Horizontal Transverse Axis)

• Standard form: $\boldsymbol{\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1}$. Branches open left and right.
• Vertices lie at $(h \pm a, k)$; the distance $a$ measures from center to each vertex.
• Asymptotes guide the branches: $y - k = \pm \frac{b}{a}(x - h)$. These lines are frequently tested.

Hyperbola (Vertical Transverse Axis)

• Standard form: $\boldsymbol{\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1}$. Branches open up and down.
• Vertices lie at $(h, k \pm a)$; the positive (first) term always indicates the opening direction.
• Asymptotes for this orientation: $y - k = \pm \frac{a}{b}(x - h)$.
• Foci relationship: $c^2 = a^2 + b^2$. Note the addition, unlike ellipses which use subtraction.

---

Measuring Shape: Eccentricity and the Focus-Directrix Definition

These concepts unify all conic sections under one framework. Eccentricity measures how much a conic deviates from being circular, and the focus-directrix definition provides a single rule that generates every conic type.

Eccentricity Formula

• Formula: $\boldsymbol{e = \frac{c}{a}}$ where $c$ is the center-to-focus distance and $a$ is the center-to-vertex distance.
• Classification by value: circle ($e = 0$), ellipse ($0 < e < 1$), parabola ($e = 1$), hyperbola ($e > 1$).
• Interpretation: higher eccentricity means a more elongated or "extreme" shape. An ellipse with $e = 0.1$ looks nearly circular, while one with $e = 0.95$ is very stretched out.

Focus-Directrix Definition

• Universal principle: a conic is the set of all points where $\frac{\text{distance to focus}}{\text{distance to directrix}} = e$.
• Parabolas have $e = 1$, meaning the distance to the focus always equals the distance to the directrix exactly.
• This definition explains why all conics share structural similarities: they're variations of the same geometric relationship, just with different ratios.

---

Quick Reference Table

---

Self-Check Questions

1. What's the key difference between the standard form equations for ellipses and hyperbolas, and how can you identify each instantly?

2. Given the equation $\frac{(x - 2)^2}{16} + \frac{(y + 3)^2}{25} = 1$, is this a horizontal or vertical ellipse? How do you know?

3. Compare how foci are calculated for ellipses versus hyperbolas. Why does one use addition and the other subtraction?

4. A parabola has vertex at $(1, -2)$ and focus at $(1, 0)$. What is the value of $p$, and what is the equation of the directrix?

5. If you're given an eccentricity of $e = 0.6$, what type of conic section is it, and what does this value tell you about its shape compared to a conic with $e = 0.9$?