Study guides › Level 7 · Geometry

The equation of a circle

Level 7 · Geometry · for questions like “A circle is (x − 3)² + (y + 2)² = 25. What are its center and radius?”

💡 The idea

(x − a)² + (y − b)² = r² is a circle with center (a, b) and radius r.

🪜 The steps

1. Read the center with the signs flipped: (x − 3) means a = 3, and (y + 2) means b = −2.
2. The radius is the square root of the number on the right.

✏️ Worked examples

1. (x − 3)² + (y + 2)² = 25: center (3, −2), radius 5.

2. (x + 1)² + (y − 4)² = 49: center (−1, 4), radius 7.

⚠️ Watch out

• The signs flip: (x + 1) means the center's x is −1.
• The right side is r², not r.

🎯 Try these

1. (x − 5)² + (y − 1)² = 9
2. (x + 2)² + (y + 6)² = 64
3. x² + (y − 3)² = 100

Answers: 1) center (5, 1), radius 3 2) center (−2, −6), radius 8 3) center (0, 3), radius 10

This guide is also in the 10MathQ app: tap “📖 Study guide” after a question.