Equations of a Circle

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Description:

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 = r²], 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.

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Tutorial

Introduction to Circles

This video presents the standard form of the equation of a circle and shows how it relates to the graph of the circle.

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Source: David on Guaranteach

Derivation of the Standard Equation of a Circle

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 $square root of left parenthesis x minus h right parenthesis squared plus left parenthesis y minus k right parenthesis squared end root$ . We know that we want this distance to be r, so we have $r space equals space square root of left parenthesis x minus h right parenthesis squared plus left parenthesis y minus k right parenthesis squared end root$ . By squaring both sides, we get $r squared equals left parenthesis x minus h right parenthesis squared plus left parenthesis y minus k right parenthesis squared$ , which is the standard form of the equation of a circle.