Equations of Circles — SAT Math Guide

Q: What You Need to Know

• **Standard form**: (x - h)² + (y - k)² = r² where (h, k) is the center and r is the radius • **General form**: x² + y² + Dx + Ey + F = 0 (can be converted to standard form) • **Completing the square** transforms general form to standard form • **Distance from center** to any point on the circle eq

Equations of circles SAT problems test your ability to work with the standard and general forms of circle equations. These questions involve finding centers, radii, and determining whether points lie on circles. You'll encounter 2-3 circle equation problems in the Geometry and Trigonometry domain on the Digital SAT. Master these concepts and you'll tackle circle problems with confidence.

What You Need to Know

Standard form: (x - h)² + (y - k)² = r² where (h, k) is the center and r is the radius

General form: x² + y² + Dx + Ey + F = 0 (can be converted to standard form)

Completing the square transforms general form to standard form

Distance from center to any point on the circle equals the radius

Diameter endpoints can help you find the center (midpoint formula)

📐 KEY FORMULA: (x - h)² + (y - k)² = r²

💡 PRO TIP: Always identify the center and radius first — most SAT circle problems become much easier once you know these values.

How to Solve Equations of Circles on the SAT

Example Question 1 — Medium Difficulty

What is the radius of the circle with equation x² + y² - 6x + 8y + 9 = 0?

A) 2

B) 4

C) 8

D) 16

Solution:

Step 1: Complete the square for x terms: x² - 6x = (x - 3)² - 9

Step 2: Complete the square for y terms: y² + 8y = (y + 4)² - 16

Step 3: Substitute and simplify: (x - 3)² - 9 + (y + 4)² - 16 + 9 = 0, so (x - 3)² + (y + 4)² = 16

✅Answer: B — The equation (x - 3)² + (y + 4)² = 16 shows r² = 16, so r = 4.

Example Question 2 — Hard Difficulty

A circle has center (2, -3) and passes through the point (6, 0). Which of the following points also lies on this circle?

A) (-2, -3)

B) (2, 2)

C) (5, -7)

D) (-1, 1)

Solution:

Step 1: Find the radius using the distance formula: r = √[(6-2)² + (0-(-3))²] = √[16 + 9] = 5

Step 2: Write the circle equation: (x - 2)² + (y + 3)² = 25

Step 3: Test each answer choice by substituting coordinates into the equation

Step 4: For choice C: (5 - 2)² + (-7 + 3)² = 9 + 16 = 25 ✓

✅Answer: C — Point (5, -7) satisfies the circle equation with center (2, -3) and radius 5.

Common SAT Math Mistakes to Avoid

❌Mistake: Forgetting to change signs when identifying center from standard form

✅Fix: Remember (x - h)² means the center x-coordinate is +h, and (x + h)² means it's -h

❌Mistake: Confusing r² with r when reading the equation

✅Fix: Always take the square root of the right side to find the actual radius

❌Mistake: Making arithmetic errors when completing the square

✅Fix: Double-check your work by expanding your completed square back out

❌Mistake: Not converting general form to standard form when finding center and radius

✅Fix: Complete the square for both x and y terms to reveal the circle's properties clearly

Practice Question — Try It Yourself

The equation of a circle is (x + 1)² + (y - 4)² = 36. What is the distance from the center of this circle to the point (5, -2)?

A) 6

B) 10

C) 12

D) 36

Show Answer

Answer: B — The center is (-1, 4), so the distance is √[(5-(-1))² + (-2-4)²] = √[36 + 36] = √72 = 6√2 ≈ 10.

Key Takeaways for the SAT

Standard form (x - h)² + (y - k)² = r² immediately shows center (h, k) and radius r

Complete the square to convert general form to standard form on the Digital SAT

Use the distance formula to find radius when given center and a point on the circle

Check if points lie on circles by substituting coordinates into the equation

SAT math equations of circles often combine with coordinate geometry and distance concepts