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## Topic D: Equations for circles and their tangents

# Features of a circle from its standard equation

CCSS.Math:

## Video transcript

The equation of a
circle C is x plus 3 squared plus y minus 4
squared is equal to 49. What are its center
h, k and its radius r? So let's just remind
ourselves what a circle is. You have some point,
let's call that h, k. The circle is the set
of all points that are equidistant from that point. So let's take the set
of all points that are, say, r away from h, k. So let's say that this
distance right over here is r, and so we want all of the
set of points that are exactly r away. So all the points x comma
y that are exactly r way. And so you could imagine
you could rotate around and all of these points are
going to be exactly r away. And I'm going to try my
best to draw at least a somewhat perfect looking circle. I won't be able to do a perfect
job of it, but you get a sense. All of these are
exactly r away, at least if I were to draw it properly. They are r away. So how do we find an equation
in terms of r and h, k, and x and y that describes
all these points? Well, we know how to find
the distance between two points on a coordinate plane. In fact, it comes straight out
of the Pythagorean theorem. If we were to draw a vertical
line right over here, that essentially is the change in the
vertical axis between these two points, up here, we're
at y, here we're k, so this distance is
going to be y minus k. We can do the exact same
thing on the horizontal axis. This x-coordinate is x while
this x-coordinate is h. So this is going to be x
minus h is this distance. And this is a right triangle,
because by definition, we're saying, hey, we're measuring
vertical distance here. We're measuring
horizontal distance here, so these two things
are perpendicular. And so from the
Pythagorean theorem, we know that this squared
plus this squared must be equal to our
distance squared, and this is where the
distance formula comes from. So we know that x minus h
squared plus y minus k squared must be equal to r squared. This is the equation
for the set-- this describes any x
and y that satisfies this equation will
sit on this circle. Now, with that out of the way,
let's go answer their question. The equation of the
circle is this thing. And this looks awfully
close to what we just wrote, we just have to
make sure that we don't get confused
with the negatives. Remember, it has to be in the
form x minus h, y minus k. So let's write it a
little bit differently. Instead of x plus
3 squared, we can write that as x minus
negative 3 squared. And then plus--
well this is already in the form-- plus
y minus 4 squared is equal to, instead of 49, we
can just call that 7 squared. And so now it
becomes pretty clear that our h is
negative 3-- I want to do that in the red color--
that our h is negative 3, and that our k is positive
4, and that our r is 7. So we could say h comma k
is equal to negative 3 comma positive 4. Make sure to get-- you
know you might say, hey, there's a
negative 4 here, no. But look, it's minus k, minus 4. So k is 4. Likewise, it's minus h. You might say, hey,
maybe h is a positive 3, but no you're subtracting the h. So you'd say minus negative 3,
and similarly, the radius is 7.