College Math Placement Practice Test

What is the standard form of the equation of a circle centered at (h, k) with radius r?

1. (x - h)² + (y - k)² = r²

2. (x + h)² + (y + k)² = r²

3. (x + h)² + (y - k)² = r²

4. (x - h)² - (y - k)² = r²

The correct answer is: (x - h)² + (y - k)² = r²

The equation of a circle in standard form is derived from the definition of a circle as the set of all points that are a fixed distance (the radius) from a given point (the center). For a circle centered at the point (h, k), the coordinates of any point on the circle are (x, y). The distance between the center (h, k) and any point (x, y) is represented by the distance formula, which must equal the radius r. This leads to the expression: \[ \sqrt{(x - h)² + (y - k)²} = r. \] To eliminate the square root, you square both sides, resulting in: \[ (x - h)² + (y - k)² = r². \] This equation captures all of the necessary components: it establishes that we are dealing with a circle, specifies the center (h, k), and incorporates the radius squared, r². None of the other options accurately reflect the structure needed for the standard form of a circle. Choices that introduce incorrect signs or placements of h and k fail to represent the fundamental geometric properties of a circle accurately.