# Standard Form of a Circle

12.3 The Circle

The next conic that we would like to examine is the circle

**Standard form of a Circle**

Let r be the radius of a circle and let (h, k) be the
center of the circle. Then the equation of the

circle is (x - h)^{2} + (y-k)^{2} =r^{2}

Graphing a circle is fairly easy. All we need to do is get
the equation into standard form and then

plot the center and use the radius to determine the rest. We will also want to
find the x-and y-

intercepts.

Example 1:

Graph the following. Find all x and y-intercepts.

Solution :

a. Since the equation is already in standard form, we can simply read off the
center of the circle.

Notice that the equation is (x - h)^{2} + (y-k)^{2} =r^{2 }
and the center is (h, k),. That means

the center is (1, -2) in our case. We can also read off the radius. Since we see
that

r^{2 }= 25 we clearly have r=5 . Now to graph, we simply plot the
center and go out from it 5

units in each direction. We get the following graph

Now we still need to find the x- and y-intercepts. We do
this the same way we always have.

For x-intercepts, set y = 0 and for y-intercepts, set x = 0.

x-intercepts

y-intercepts:

Looking at the graph again we can see clearly these are
the correct values.

b. This time we are not given the equation in such a nice form. We will need to
put the equation

into standard form in order to get the important information. To do that, we
will complete the

square just like in the previous section . However, this time, since we want to
have the radius

on the right hand side, we will add our completed square portion to both sides
instead of

adding it in and then subtracting it off of the same side as we did with the
parabola. We

proceed as follows

Now we can clearly see that the center is (-4,1)− and the
radius is r=2.

So we graph the circle by plotting the center and going out 2 units in each
direction. We

get

Lastly we need to find the x- and y-intercepts.

We can see from the graph that we should have two x -intercepts and no
y-intercepts

since it hits the x-axis twice and does not hit the y-axis at all. Lets verify
this with usual

steps .

x-intercepts:

y-intercepts:

So we can see that since we get imaginary values for our
y -intercepts, we have no y-

intercepts, just as our graph had indicated.

One of the main things that we want to do in this chapter is to be able to
distinguish between the

various conics from the very beginning of the problem. So far the only conics
that we have are

the parabola and circle . It is very easy to tell a parabola from all the other
conics, since the

parabola will be the only one that only has one variable that is squared .

So, at the moment we have the following

Only one of the two variables is squared- the conic is a parabola

Both variables are squared- the conic is a circle

**12.3 Exercises**

Graph the following. Identify the center, radius and all x and y-intercepts.

Identify if the following is a parabola or a circle. Then
graph accordingly.

Find the equation of the circle that satisfies the given
conditions.

35. center (-1, 3), r =3

36. center (2,-5), r=4

37. center (−1, 1)and point(3,4) is on the circle

38. center(−2,-3) and point (1,1) is on the circle

39. (-1,1)and (3,4)are endpoints of the diameter

40. (-2.-3)and (1,1)are endpoints of the diameter

41. center ( -2,1)and one y- intercept is at -3

42. center (4,-5) and one x-intercept is at 3

Find the area of the following circles.

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