Conic Sections Equations- Complete Guide with Examples

What Are Conic Sections?

Conic sections are the curves you get when a plane cuts through a cone. Circle, ellipse, parabola, hyperbola—these are the four shapes you need to know. Each has its own equation, its own personality, and its own set of problems on exams.

You probably first encountered these in algebra or precalculus. Maybe you're revisiting them for calculus or physics. Either way, the equations follow patterns. Once you see the patterns, everything clicks.

The Four Conic Sections at a Glance

Before diving into equations, here's the quick rundown:

• Circle — All points equidistant from one center
• Ellipse — Sum of distances from two foci stays constant
• Parabola — All points equidistant from one focus and a directrix
• Hyperbola — Difference of distances from two foci stays constant

Each one appears in real-world applications. Satellites use elliptical orbits. Telescopes use parabolic mirrors. Bridge cables follow hyperbolic curves. But let's focus on the math first.

Circle Equations

The circle is the simplest conic section. No ovals, no open curves—just a perfect loop.

Standard Form (Center at Origin)

x² + y² = r²

Here, r is the radius. If you see x² + y² = 25, the radius is 5. That's it.

Standard Form (Center at (h, k))

(x - h)² + (y - k)² = r²

The center shifts from the origin. The equation (x - 3)² + (y + 2)² = 16 has center (3, -2) and radius 4.

Watch the signs. It's (x - h) and (y - k), not (x + h). If your center is (3, -2), then h = 3 and k = -2, giving you (x - 3) and (y - (-2)) = (y + 2). Don't trip over the negatives.

General Form

x² + y² + Dx + Ey + F = 0

You can convert general form to standard form by completing the square. That's a skill you'll need repeatedly.

Ellipse Equations

An ellipse is a stretched circle. It has two axes—a longer one (major axis) and a shorter one (minor axis).

Standard Form (Center at Origin)

Horizontal major axis: x²/a² + y²/b² = 1

Vertical major axis: x²/b² + y²/a² = 1

The larger denominator is always a². That tells you the major axis direction. If a² is under x², the ellipse stretches left and right. If a² is under y², it stretches up and down.

Standard Form (Center at (h, k))

Horizontal: (x - h)²/a² + (y - k)²/b² = 1

Vertical: (x - h)²/b² + (y - k)²/a² = 1

Foci of an Ellipse

The foci sit along the major axis. For a horizontal ellipse, they're at (c, 0) and (-c, 0). For vertical, they're at (0, c) and (0, -c).

The relationship is c² = a² - b². Always. Remember this formula—it's the one you'll use most.

Parabola Equations

A parabola opens in one direction. It has a vertex and a focus, plus a directrix line on the opposite side.

Standard Form (Vertex at Origin)

Opens right: y² = 4ax

Opens left: y² = -4ax

Opens up: x² = 4ay

Opens down: x² = -4ay

The sign of the constant tells you the direction. Positive opens toward positive axis; negative opens toward negative axis.

Standard Form (Vertex at (h, k))

Opens right: (y - k)² = 4a(x - h)

Opens left: (y - k)² = -4a(x - h)

Opens up: (x - h)² = 4a(y - k)

Opens down: (x - h)² = -4a(y - k)

The focus sits inside the parabola, a units from the vertex in the direction it opens. The directrix is a units on the other side.

Hyperbola Equations

A hyperbola has two separate branches. It opens either left-right or up-down, depending on which term is positive.

Standard Form (Center at Origin)

Opens left-right: x²/a² - y²/b² = 1

Opens up-down: y²/a² - x²/b² = 1

The positive term tells you the direction. If x² is positive, the hyperbola opens left and right. If y² is positive, it opens up and down.

Standard Form (Center at (h, k))

Opens left-right: (x - h)²/a² - (y - k)²/b² = 1

Opens up-down: (y - k)²/a² - (x - h)²/b² = 1

Foci of a Hyperbola

For hyperbolas, the foci are farther out than the vertices. The relationship is c² = a² + b². Notice this is different from ellipses—you add here instead of subtract.

Comparison: Conic Section Equations

This table is worth memorizing. When you see an equation, you should immediately recognize which conic it represents.

How to Identify a Conic from Its Equation

Look at the general second-degree equation:

Ax² + Bxy + Cy² + Dx + Ey + F = 0

The discriminant B² - 4AC tells you what you're dealing with:

• B² - 4AC < 0 → Ellipse (or circle if A = C and B = 0)
• B² - 4AC = 0 → Parabola
• B² - 4AC > 0 → Hyperbola

If B = 0 and A = C, you have a circle. No need to calculate further.

Getting Started: Solving Conic Section Problems

Here's the process that works every time:

Step 1: Identify the Conic

Look at the equation structure. Does it have one squared term? Two squared terms with the same signs? Two squared terms with opposite signs? Each pattern points to a specific conic.

Step 2: Write the Standard Form

Complete the square if needed. Move constant terms, group x's, group y's, and factor out coefficients. The goal is to get the equation into its standard form so you can read off the important parts.

Step 3: Extract Key Information

Once in standard form, you can immediately identify:

• Center or vertex
• Orientation (opens which direction)
• Key distances (a, b, c, r)
• Foci locations

Example: Converting to Standard Form

Given: x² + 4x + y² - 6y = 12

Group and complete the square:

(x² + 4x) + (y² - 6y) = 12

(x² + 4x + 4) + (y² - 6y + 9) = 12 + 4 + 9

(x + 2)² + (y - 3)² = 25

This is a circle. Center at (-2, 3), radius = 5. Done.

Common Mistakes to Avoid

• Confusing ellipse and hyperbola formulas — Ellipse: c² = a² - b². Hyperbola: c² = a² + b². Write these down until they stick.
• Wrong sign on vertex coordinates — In (x - h)², the center is at +h. If your equation has (x + 2)², that's actually (x - (-2))², so h = -2.
• Forgetting to complete the square — You can't read off the center or radius from the general form. Always convert to standard form first.
• Mixing up a and b in ellipses — a is always the larger denominator. It tells you the major axis. If you're not sure, check which denominator is bigger.

Quick Reference: The Formulas You Need

• Circle: (x - h)² + (y - k)² = r²
• Ellipse: (x - h)²/a² + (y - k)²/b² = 1 (or swapped)
• Parabola: (x - h)² = 4a(y - k) or (y - k)² = 4a(x - h)
• Hyperbola: (x - h)²/a² - (y - k)²/b² = 1 (or swapped)

Memorize these. They're the foundation for everything else you'll do with conic sections.