Example 1: Circle Centered at Origin

Problem: Write the equation of a circle centered at (0, 0) with radius 4.

Solution: Since h = 0 and k = 0, the equation simplifies to: x² + y² = 16.

This is the simplest form of a circle equation, where the center is at the origin of the coordinate system.

Example 2: Finding the Center and Radius

Problem: Find the center and radius of the circle: (x + 4)² + (y - 1)² = 36

Solution: Comparing to (x - h)² + (y - k)² = r², we have (x - (-4))² + (y - 1)² = 6².
Therefore, the center is (-4, 1) and the radius is 6.

Remember that (x + 4) means (x - (-4)), so h = -4, not +4.

Example 3: Converting from General to Standard Form

Problem: Convert x² + y² - 6x + 8y - 11 = 0 to standard form.

Solution: Group x and y terms: (x² - 6x) + (y² + 8y) = 11.
Complete the square for each: (x² - 6x + 9) + (y² + 8y + 16) = 11 + 9 + 16.
This gives: (x - 3)² + (y + 4)² = 36, so center is (3, -4) and radius is 6.

Example 4: Equation from Three Points

Problem: Three points on a circle are (0, 0), (4, 0), and (0, 4). Find the circle's equation.

Solution: The center must be equidistant from all three points. By symmetry and calculation, the center is (2, 2). The distance from (2, 2) to (0, 0) is √[(2-0)² + (2-0)²] = √8 = 2√2.
The equation is: (x - 2)² + (y - 2)² = 8.

Watch Your Signs

In the equation (x - h)² + (y - k)² = r², the center coordinates have opposite signs from what appears in the equation. If you see (x + 3), the h-value is -3. If you see (y - 5), the k-value is +5. This is one of the most common sources of errors when identifying centers.

Remember r² vs r

The right side of the standard form equation is r², not r. If you see (x - 2)² + (y + 1)² = 49, the radius is 7, not 49. Always take the square root of the right side to find the actual radius. This distinction is crucial for accurate calculations.

Completing the Square Carefully

When converting from general form to standard form, complete the square for x and y separately. Take half of the coefficient of the linear term, square it, and add it to both sides. For x² + 6x, take (6/2)² = 9. This gives you (x + 3)². Don't forget to add the same values to the other side of the equation.

Verify Your Answer

After finding a circle's equation, verify it by checking that known points on the circle satisfy the equation. Substitute the coordinates into your equation and confirm that both sides equal. This quick check catches calculation errors before they become bigger problems.

Center

The fixed point (h, k) that is equidistant from all points on the circle. The center defines the circle's position in the coordinate plane and is a fundamental parameter in the circle's equation.

Radius

The constant distance r from the center to any point on the circle. The radius determines the circle's size and appears as r² in the standard form equation. A larger radius means a larger circle.

Standard Form

The equation (x - h)² + (y - k)² = r² that explicitly shows the center (h, k) and radius r of a circle. This form is most useful for quickly identifying a circle's properties and for graphing.

General Form

The expanded form of a circle's equation: x² + y² + Dx + Ey + F = 0. While less intuitive than standard form, it's algebraically equivalent and useful for certain calculations and when solving systems of equations.

Completing the Square

An algebraic technique used to convert the general form of a circle's equation to standard form. The process involves creating perfect square trinomials from quadratic expressions, revealing the circle's center and radius.

Diameter

A line segment that passes through the center and has its endpoints on the circle. The diameter equals twice the radius (d = 2r) and represents the longest distance across the circle.