Circles Page 10b

Page 10b

Exploring Another Special Form of the Circle Equation

You just saw that the standard equation x² + y² = P represents a circle when P > 0.

Now let's look at the equation x² + y² + F = 0.
Use the action figure to explore the following question:
For what values of F does the equation x² + y² + F = 0 describe a circle?
Once you've explored on your own, answer the questions in the question box to check your understanding of the effect of F in the equation x² + y² + F = 0.

x² + y² + F = 0

In summary:
x² + y² = P represents a circle when P > 0.
x² + y² + F = 0 represents a circle when F < 0.

Note that x² + y² + F = 0 can be rewritten as x² + y² = -F.
Relating back to the equation x² + y² = P, we notice that P = -F.

Of course this is the case because if P > 0 then -F > 0, which implies F < 0.

In summary:	x² + y² = P represents a circle when P > 0. x² + y² + F = 0 represents a circle when F < 0.
Note that x² + y² + F = 0 can be rewritten as x² + y² = -F. Relating back to the equation x² + y² = P, we notice that P = -F.
Of course this is the case because if P > 0 then -F > 0, which implies F < 0.