Finding Circle Equations- Methods and Examples

What Is a Circle Equation?

A circle equation tells you the relationship between every point on a circle and its center. That's it. If you know the center coordinates and the radius, you can write the equation. If you have the equation, you can find the center and radius.

The two forms you need to know:

• Standard form: (x - h)² + (y - k)² = r²
• General form: x² + y² + Dx + Ey + F = 0

h and k are the center coordinates. r is the radius. Everything else is just rearranging these basic pieces.

Standard Form: The One You Will Use Most

The standard form is (x - h)² + (y - k)² = r².

Here's what each part means:

• h = x-coordinate of the center
• k = y-coordinate of the center
• r = radius (distance from center to any point on the circle)

Example: A circle with center at (3, -2) and radius 5.

Plug in the values: (x - 3)² + (y - (-2))² = 5²

Simplify: (x - 3)² + (y + 2)² = 25

That's your answer. You just substitute the numbers and you're done.

Common Mistake to Avoid

Watch the signs. The formula has (x - h), not (x + h). If your center is at (-3, 4), then h = -3, so you write (x - (-3)) = (x + 3). The math does not lie. Check your signs every time.

General Form: When You Need to Complete the Square

The general form looks messy: x² + y² + Dx + Ey + F = 0

You cannot read the center and radius directly from this form. You have to convert it to standard form by completing the square.

Let's convert x² + y² - 4x + 6y - 3 = 0 to standard form.

Step 1: Group x-terms and y-terms

(x² - 4x) + (y² + 6y) = 3

Step 2: Complete the square for each group

For x² - 4x: take half of -4 (which is -2), square it (4), add it inside the parentheses.

For y² + 6y: take half of 6 (which is 3), square it (9), add it inside the parentheses.

(x² - 4x + 4) + (y² + 6y + 9) = 3 + 4 + 9

Step 3: Write as perfect squares

(x - 2)² + (y + 3)² = 16

Center is (2, -3). Radius is 4.

Finding the Equation from Different Given Information

From the Center and a Point on the Circle

You have the center (h, k). You have one point (x₁, y₁) that lies on the circle. The radius is the distance between them.

Use the distance formula: r = √[(x₁ - h)² + (y₁ - k)²]

Then plug everything into (x - h)² + (y - k)² = r²

Example: Center at (1, 2). Point on circle is (4, 6).

r = √[(4-1)² + (6-2)²] = √[9 + 16] = √25 = 5

Equation: (x - 1)² + (y - 2)² = 25

From Three Points on the Circle

Three points determine a unique circle. You set up a system of equations.

Start with the general form: x² + y² + Dx + Ey + F = 0

Plug in each point's coordinates. You get three equations with three unknowns (D, E, F). Solve the system.

This gets messy fast. Use a calculator or matrix method if you can. The algebra is not worth doing by hand when numbers get ugly.

From the Diameter

If you know the endpoints of a diameter, the center is the midpoint. The radius is half the distance between the endpoints.

Endpoints: (x₁, y₁) and (x₂, y₂)

Center: ((x₁+x₂)/2, (y₁+y₂)/2)

Radius: ½ × √[(x₂-x₁)² + (y₂-y₁)²]

Then write the standard form equation with these values.

Practical How To: Step-by-Step Process

When you face a circle equation problem, follow this order:

1. Identify what you know. Center? Radius? Points? Diameter?
2. Choose the right approach. Direct substitution, distance formula, or midpoint formula.
3. Calculate what you need. Radius from distance, center from midpoint, or vice versa.
4. Write the standard form. (x - h)² + (y - k)² = r²
5. Expand to general form if asked. Move everything to one side, simplify.

Quick Reference Table

Converting Between Forms

Sometimes you need to switch from standard to general form, or the other way around.

Standard to General:

Start with (x - 3)² + (y + 1)² = 16

Expand: x² - 6x + 9 + y² + 2y + 1 = 16

Combine: x² + y² - 6x + 2y + 10 - 16 = 0

Simplify: x² + y² - 6x + 2y - 6 = 0

General to Standard:

Use completing the square, as shown earlier. Group x-terms, group y-terms, add/subtract to make perfect squares.

Real Example: Putting It All Together

Problem: Find the equation of a circle passing through (2, 3) and (5, 7) with center on the line x - 2y = 3.

This one requires more work. The center lies on a line, so you do not know exact coordinates yet. You set up two conditions:

• Distance from center to (2,3) equals distance from center to (5,7)
• Center satisfies x - 2y = 3

Let center be (h, k). From the line: h - 2k = 3, so h = 2k + 3.

From equal distances: (h-2)² + (k-3)² = (h-5)² + (k-7)²

Expand and simplify. You get a linear relationship between h and k.

Solve the system with h = 2k + 3. You get the center. Then find the radius from one of the points. Then write the equation.

These problems take time. Do not rush. Write down every step.

Common Errors That Cost Points

• Forgetting to square the radius when writing the equation
• Getting the sign wrong when the center has negative coordinates
• Not completing the square correctly in the general form
• Using the diameter endpoints as center instead of midpoint
• Mixing up x and y coordinates when calculating distance

Double-check each of these before you submit an answer.

When to Use Which Form

Use standard form when you have the center and radius. It is clean and shows the center immediately.

Use general form when the problem gives you the equation in that format, or when you need to identify circle properties from a given equation.

Convert between them as needed. The problem will tell you which form it wants, or you figure it out from context.