Circle equations — Digital SAT Math practice
Standard form, center and radius, completing the square.
Key terms
- Center: The fixed point (h, k) equidistant from all points on the circle.
- Radius: The constant distance r from the center to any point on the circle.
- Standard Form: The equation format showing center and radius explicitly.
- General Form: The expanded quadratic format x^2 + y^2 + Dx + Ey + F = 0 requiring completion of squares.
- Completing the Square: Algebraic method to convert general form into standard form by adding constant terms.
Common traps
- Forgetting to divide the linear coefficient by two before squaring it when completing the square.
- Misreading the center coordinates by ignoring the sign flip in the standard form parentheses.
- Squaring the radius value on the right side instead of taking its square root to find r.
Key ideas to know
Identify center and radius from standard form
What are the center and radius of the circle with equation (x – 4)2 + (y + 5)2 = 36?
Center is (4, –5) and radius is 6. The standard form is (x – h)2 + (y – k)2 = r2, so h = 4, k = –5, and r2 = 36.
Watch out: Forgetting that signs inside the parentheses flip when reading the center coordinates.
Convert general circle equation to standard form
Rewrite x2 + y2 – 6x + 8y = 0 in standard form to find the center and radius.
Group x and y terms, complete the square for each, and balance the equation. Result: (x – 3)2 + (y + 4)2 = 25. Center is (3, –4), radius is 5.
Watch out: Forgetting to add the completed-square constants to BOTH sides of the equation when balancing.
Radius calculation error in circle equations
A student solves (x – 2)2 + (y + 3)2 = 49 and concludes the radius is 49. What is wrong, and what is correct?
The right side equals r2, not r. The student forgot to take the square root. The correct radius is √49 = 7.
Watch out: Confusing the squared radius on the right side with the actual radius value.
Sample questions
Question 1
What is the distance between the points (1, 2) and (4, 6) in the coordinate plane?
- 5
- 7
- 25
- 3
Show answer and explanation
Answer: 5
Distance = sqrt((4 - 1)^2 + (6 - 2)^2) = sqrt(9 + 16) = sqrt(25) = 5.
Question 2
The equation x² + y² + 16x + 14y + 104 = 0 represents a circle in the xy-plane. What is the radius of the circle?
- 9
- 4
- 3
- 15
Show answer and explanation
Answer: 3
Complete the square: (x + 8)² + (y + 7)² = 9. Radius = √9 = 3.