Find the center (h, k) and radius r of each circle below.
Then, graph each circle.
(1) x^2 + (y - 1)^2 = 1
(2) x^2 + y^2 + x + y - (1/2) = 0
(3) How do I graph such circles WITHOUT a graphing calculator?
The general equation for a circle is where the center of the circle is (h, k) and the circle has a radius of r.
I won't even bother with the first one...
2)
We need to complete the square for both the x and y variables:
Now, . So looking at the first set of parenthesis we've got . Equating the linear coefficient of this expression and the one right above we see that , so . This means we want to add to to make it a perfect square since . But what we do to one side of the equation we need to do to the other. Thus:
Now to the y variable. The problem is identical and we find we need to add to both sides again:
So this circle has a center at and a radius of 1.
-Dan
a is just a "dummy variable." In order to complete the square for we need to add some constant to it. So I am comparing
<-- This is the form for the perfect square .
where is the number we need to add to . By comparing the two expressions we can see that we need the linear x term to have the same coefficient in both lines, so .
Hopefully that explains it well enough. If not I can provide other examples.
-Dan