Algebra Practice Test

The standard form of the equation of a circle is derived from the definition of a circle as the set of all points that are a fixed distance (the radius) from a central point (the center of the circle). For a circle centered at (h, k) with radius r, the formula is structured to directly showcase this geometric relationship.

The correct form is expressed as \( (x - h)² + (y - k)² = r² \). Here’s why this format works:

1. **Center Coordinates**: The terms \( (x - h) \) and \( (y - k) \) account for the horizontal and vertical distances from any point (x, y) on the circle to the circle's center (h, k). The subtraction indicates that if you are at the center (h, k) and moving horizontally to the right (for positive h) or left (for negative h), or vertically up (for positive k) or down (for negative k), it correctly represents these movements.

2. **Radius**: The equality to \( r² \) indicates that the sum of the squares of these distances must equal the square of the radius, perfectly matching the distance formula (the Pythag