Show how the Standard Equation of a Circle is derived from the Distance Formula.
Present how to write the standard form of a circle’s equation given the central point (h, k) and a radius.
Show how to find the center (h, k) and radius of a circle when given an equation in standard form, and then show how to graph the circle.
Introduce the General Form of the Equation of a Circle [(x – h)^{2} + (y – k)^{2} = r^{2}], and show how it can be derived from the Standard Form.
Convert the general form of a circle’s equation to standard form.
This packet should help a learner to understand how to use the equations of a circle to gather information and graph a circle.
This video presents the standard form of the equation of a circle and shows how it relates to the graph of the circle.
Source: David on Guaranteach
Using the distance formula, it's fairly easy to see where our standard equation of a circle comes from. We want to find all the points that are a distance r from the center point - that's the definition of a circle. Call the center point (h,k), and choose an arbitrary point (x,y) on the circle, like so:
For every (x,y), the distance formula tells us that the distance between (h,k) and (x,y) is . We know that we want this distance to be r, so we have . By squaring both sides, we get , which is the standard form of the equation of a circle.
This video demonstrates how to convert a circle's equation to standard form.
Source: Kevin Kriescher on Guaranteach
This video demonstrates how to use a graph of a circle to determine the circle's equation.
Source: David on Guaranteach
This video demonstrates how to sketch a graph of a circle.
Source: Timothy Palma on Guaranteach