Completing the Square for Circle Equations

Lesson 2 of 2: General Form and Applications

In this lesson:

• Recognize and convert circle equations from general form to standard form
• Complete the square to find the center and radius
• Apply circle equations to real-world problems

Learning Objectives

By the end of this lesson, you should be able to:

1. Derive the equation of a circle using the Pythagorean Theorem (review)
2. Write the equation in standard form (review)
3. Identify center and radius from standard form (review)
4. Complete the square to convert general form to standard form (new)
5. Find center and radius of a circle given by an equation in general form (new)
6. Explain why the equation represents all points at distance from (review)

The Problem with General Form

In Lesson 1, we had nice standard form equations:

In real problems, you might see:

Where's the center? Where's the radius?

Goal: Convert general form → standard form using completing the square

General Form vs. Standard Form

General form:

• Compact, but center and radius are hidden

Standard form:

• Center and radius are visible

Strategy: Complete the square to reconstruct the squared binomials

Completing the Square: The Rule

Starting with , we want to write it as a perfect square.

The rule: Add to both sides

Example:

Half the coefficient of , then square it.

Quick Check: What Do You Add?

To complete the square for , what number do you add?

Half the coefficient of , then square it...

Quick Check — Answer

: half of is , and

The same rule applies to terms:

: half of is , and →

Complete the square separately for terms and terms

Full Example: Step 1

Given:

Step 1: Move the constant to the right. Group and terms.

Full Example: Steps 2 and 3

Step 2: Complete the square for — add to both sides

Step 3: Complete the square for — add to both sides

Full Example: Step 4 — The Result

Step 4: Factor and read

Center:

Note: means — the sign trap applies here too

Radius:

Check: ✓

Your Turn: Try the Process

Equation:

Step 1: Group and move constant:

$(x^2 + 6x \phantom{{}+ 9}) + (y^2 - 10y \phantom{{}+ 25}) = $ ___

Step 2: Complete for : add $\left(\dfrac{6}{2}\right)^2 = $ ___

Step 3: Complete for : add $\left(\dfrac{-10}{2}\right)^2 = $ ___

Step 4: Factor and identify center and radius

Your Turn — Answer

Add and to both sides:

Center: — note: means

Radius:

Edge Cases: Always Check!

After completing the square, always check the right side:

• : real circle ✓
• : single point — only the center satisfies the equation
• : no real solution — no circle exists in the real plane

From Equations to Applications

The full toolkit:

• Standard form → read center and radius immediately
• General form → complete the square → standard form → read

Now let's apply this toolkit to real problems

Problem 1: Find the Equation

A circle passes through and has center . Write the equation.

Step 1: Find the radius using the distance formula

Step 2: Write the equation

Problem 2: General Form to Center and Radius

Find the center and radius of

(We solved this in the guided practice!)

Center: Radius:

Strategy: complete the square first, then read

Problem 3: Verify a Point

Is on the circle ?

Substitute , :

Yes — is on the circle.

The legs are 4 and 3, hypotenuse is 5 — a 3-4-5 right triangle!

Problem 4: Circle at the Origin

A circle is centered at the origin and passes through .

Radius: Distance from origin to :

Equation:

(Another Pythagorean triple: 6-8-10)

Real-World Application

Scenario: A cell tower is at coordinates on a city map (in km). Coverage radius is km.

Coverage boundary equation:

Is a house at within coverage?

No — the house is outside the coverage area.

Practice: Mixed Problems

Solve each problem:

1. Complete the square: . Find center and radius.

2. Write the equation of a circle with center passing through .

3. A radar system covers a circle centered at with radius . Is within range?

Work through all three before advancing

Practice — Answers

1. 
   → Center , radius

2. = distance from to =
   

3. 
   Radius squared: . Since ✓ Within range.

Key Takeaways

✓ General form hides center and radius

✓ Complete the square to convert to standard form — 4 steps

✓ Completing the square rule: half the coefficient, then square it

✓ Add to both sides when completing the square — balance the equation

✓ Always check that after completing the square

Watch out: Add , not — use half, not the full coefficient

Watch out: Add to both sides — one-sided additions break the equation

Watch out: Check — not every equation gives a real circle

Next: Equations of Parabolas

You've mastered circle equations — standard form, general form, and applications

Coming up: HSG.GPE.A.2

• Derive the equation of a parabola using the same geometric approach
• A parabola is defined by points equidistant from a focus and a directrix
• The derivation strategy you learned for circles carries directly over