Another Way to Think About Equations of Circles

5. Equation of Unit Circle at the Origin

Let (x'',y'') represent any point on the unit circle centered at the origin. Write an equation showing the relationship between x'' and y''.

6. Relationship Between Original Circle and Unit Circle at Origin

Think (or look) back to when you transformed the circle in the graph into the unit circle centered at the origin. What were the algebraic expressions relating the points of the transformed image (x'',y'') to their pre-images (x,y) on the original circle?

7. Transformations Approach to Equation of Any Circle

Substitute your expressions for x'' and y'' from Question 6 into your equation for the unit circle at the origin from Question 5. What is the resulting equation for the circle in the graph?

8. Is it Equivalent?

Explain how to algebraically manipulate the equation you found in Question 7 to look like the equations for circles we found before (by squaring the distance from the Pythagorean distance formula). Convince yourself that the two approaches to deriving the equation of a circle yield equivalent equations!