Equation of a Circle - Concept
A circle can't be represented by a function, as proved by the vertical line test. However, we can obtain an equation that describes the full circle by using the distance formula between the given center coordinates and any point along the circumference of the circle. Once we have derived this equation of a circle, we can apply it to any other circle you may come across in a coordinate plane.
In geometry it's helpful to come up with
an equation that will describe a
circle that's centered at some point
H and K with a radius R.
Well, before we come up with this,
let's do a little backtrack.
How can I calculate the distance
of R from point HK to XY?
Well, we said our distance formula between
any two points A and B is equal to
the square root of the differences of
the Xs squared plus the differences
of the Ys squared.
So let's apply that to this problem.
AB is actually the radius of this problem.
That's what we're trying to find.
So the radius is equal to the square root.
If I subtract my Xs, I see that I have X
as this point and my center of my circle
is at H. So we're going to say this
is X minus H quantity squared.
Now let's look at our Ys.
Our Y is Y, excuse me, our Y2 is Y. And
I'm going to subtract K. Because
K is the Y coordinate of
the center of my circle.
And I'm going to square that.
So for any point on this circle, what I'm
going to do is I'm going to square
both sides of this equation.
So that will give us any point
above or below that X axis.
Because right now if we just look at R,
we're going to be given half of this circle.
Because if you remember from algebra,
your vertical line test would fail of
the circle. The vertical line test, remember, says if
you can draw a vertical line anywhere
on your graph and it intersects your
function more than once it's not a function.
So we know a circle is not a function.
So we're going to square both sides
so we get the full function.
So R squared is equal to X minus H quantity
squared plus Y minus K quantity squared.
So the equation of a circle with radius
R is this equation right here.
Where H is the X coordinate of your center
and K is your Y coordinate of your
So you're going to be substituting in for
HK and for R and that will give you
the equation of your circle.