Welcome to our lesson on the standard equation of a circle. A circle is defined as the set of all points that are equidistant from a fixed center point. This distance is called the radius. Let's visualize this concept with a circle centered at point h, k with radius r.
Now let's derive the standard equation. Consider any point x, y on the circle. The distance from this point to the center h, k is given by the distance formula: square root of x minus h squared plus y minus k squared. Since this point is on the circle, this distance equals the radius r. Squaring both sides gives us the standard equation: x minus h squared plus y minus k squared equals r squared.
Let's look at a special case: a circle centered at the origin. When the center is at zero, zero, the standard equation simplifies significantly. The h and k values are both zero, so our equation becomes x squared plus y squared equals r squared. For example, a circle with radius 2 centered at the origin has the equation x squared plus y squared equals 4. This is the simplest form of a circle equation.
Now let's work through a complete example. Consider a circle with center at 2, negative 1 and radius 3. We substitute these values into our standard equation. The center coordinates h equals 2 and k equals negative 1, and the radius r equals 3. Substituting into the formula gives us x minus 2 squared plus y minus negative 1 squared equals 3 squared. Simplifying, we get x minus 2 squared plus y plus 1 squared equals 9. This is the standard equation for our circle.
To summarize, the standard equation of a circle is x minus h squared plus y minus k squared equals r squared, where h and k are the center coordinates and r is the radius. This fundamental equation has wide applications in geometry, engineering, physics, computer graphics, and navigation systems. Understanding this equation allows us to describe any circle precisely using just three parameters: the x-coordinate of the center, the y-coordinate of the center, and the radius.