Circle Equations Explained with Simple Animations

Let's explore the standard form of a circle equation.The standard form equation describes any circle using three key values: h, k, and r.Let's start with a circle centered at the origin, with radius 2.In this equation, h represents horizontal shift, k represents vertical shift, and r is the radius.Let's look at a practical example. This circle has its center at (2,3) and a radius of 2.We can verify points on the circle. For example, the point (4,3) lies exactly 2 units from the center.To convert from general form to standard form, we'll use the method of completing the square.First, let's group the x terms and y terms separately.For the x terms, we complete the square by taking negative D over 2, which is 2, and squaring it.The same process applies to the y terms, where we take negative E over 2, which is negative 3.Let's understand how we got these values. For x terms, negative D over 2 gives us h, which is 2.For y terms, negative E over 2 gives us k, which is negative 3.These formulas help us quickly convert between general and standard form.The constant term on the right side becomes r squared, giving us our radius.To graph a circle, we'll use the equation (x minus 2) squared plus (y plus 1) squared equals 9.Let's follow a step-by-step process to graph this circle.First, we identify the center point. From the equation, h equals 2 and k equals negative 1.Next, we determine the radius. The right side of the equation is 9, so the radius is 3 units.Now we can draw our circle with center at (2, negative 1) and radius 3 units.Let's verify if the point (4, 0) lies on our circle by plugging it into the equation.When we substitute the point, we get 5, which is not equal to 9. Therefore, this point does not lie on the circle.Let's look at a practical application. If we want to design a circular garden with radius 3 meters...The garden would have an area of approximately twenty-eight point two seven square meters.Let's review what we've learned about graphing circles.Thanks for learning about circles with Spark.E!