Standard Form of the Equation of a Circle (Fundamentals)

Video Series: Functions and Their Graphs (Step-by-Step Tutorials for Precalculus)

(D4M) — Here is the video transcript for this video.

00:00
in this episode you’ll learn how to write an equation of a circle in
standard form and sketch its graph [Music] okay so we’re going to start off by
looking at what the standard form of the
equation of the circle is so we’re going
to take an arbitrary point on the circle so let’s just draw a circle right here
and my circles won’t be perfect but that’s good enough so we’re going to
take an arbitrary point here x and y on the circle
and our center right here is going to be h k and we’re going to have a radius of
length r and so we’re going to say that this is
the standard form for the equation of the circle
uh this is an arbitrary point here x y you can see it better and
the center is h k and so this will be our equation right here

00:01
so this comes from the pythagorean theorem and the distance formula
and we did a video on those already so a point is on the circle all the
points that are equal distant from the center right and so that’s the the equal
distance that’s the radius right there so let’s look at some examples
so for our first example right here we’re going to look at
the center is at zero zero and the radius is four so the first example is
so the h and the k are zero so this is going to be x minus zero
in other words we’re just going to get x square plus y squared equals and then
the radius is four so we’re going to get four squared is 16 and so our circle
will just look like uh that one’s a little bit too [Music]
so our circle looks something like this this will be a four and a four

00:02
and a minus four and a four and so you know that’s just the circle there
for number two for part two um here now we’ve moved the circle away
from the origin and we’re at minus seven minus four radius seven
so here’s what that equation would look like it would look like something like
x minus so because it’s a minus here and then
our h whatever our h is our h is a minus seven so i’m gonna get a minus minus
seven so i’m gonna say plus seven and then y minus so it’s right y minus
and then the k and the k is also a minus so another minus minus
and then equals and then our radius is 7 so it’s going to be equal to 49 here
and so let’s see if we can get a good sketch of this graph here
so the circle will look something like maybe i should try to draw the circle

00:03
first all right so let’s say there’s the circle there and it’s gonna
hit right there and come through here x y and this will be the center right here
at minus seven and minus four so there’s a center right there and
there’s the equation for this circle right there so this distance is seven
away from the origin so it’s minus seven and that’s what we get right there
and then minus four it goes down minus four so so
it’s right here about minus four down and it touches right there i know
because that distance is seven right there okay so
there would be the uh number two there what about number three so number three

00:04
the end points of the diameter so let’s try to sketch this graph here
we got a circle going on right here and come through here
and i’m also going to dash this line right here
um this this is the line coming through here through the origin
and this equation is for this line right here is its um 4x
um when no it’s one fourth x so when x is four then the y is one
when the x is minus one minus four then we get a minus one out
um so this would be the point right here uh on the circle four one
and then this would be the point right here minus four minus one and the center

00:05
right here would be at zero zero uh another way to check that out is by
finding the midpoint of the diameter and so if you just add up the fours
and then divide by two you get zero and add up the ones and divide by two you
get zero so here’s gonna be the midpoint right here
um so we’re gonna be looking like x squared because it’s gonna go through
the origin here x squared plus y squared equals and now what is the radius so
let’s find the distance between these two points here so the distance is
square root of and so this is going to be um so let’s
look at the x’s here so it’s going to be 4 plus four squared plus and then
one minus a negative one so one plus one squared
and so that distance is going to be eight squared so 64 plus 4
so that’ll be square root of 68 which i’ll reduce to 2 square roots of 17

00:06
and let’s let’s make sure you can see me there so this will be 8 square so 64
plus 2 squared 2 square roots of 17 and so the diameter that’s the diameter
so the radius will be half of it so the radius is square root of 17
and so our equation will be r squared so square root of 17 squared so that’ll be
just 17 there so there’s our um equations in standard form
for number three there so let’s see about trying to find
uh equation in standard form if you have some x’s and y’s because here’s got
perfect squares in here so let’s look at this next part here
completing the square here now if you’ve never seen completing the
square before well i’m going to go slow here on this one in case you haven’t
seen that and you’ll look at it in more detail later but let’s see what we can

00:07
do right now so it’s number one here we have an x squared and what i’m going
to do is i’m going to pull that this x along next to it so i’m just
going to write it right here and then we have a y squared so plus y squared
and then i’m going to say 28 y and then i’m going to put the 181 on
this other side over here so negative 181 so i pulled the x’s together and i
pulled the y’s together and so what i’m going to do is i’m going
to look at these two terms here and i’m going to look at these two terms
here and i’m going to do something separately but i’m going to think about
these two first so what i’m going to do is i’m going to
look at this number in front of the x i’m going to take half of it and i’m
going to square it so i’m going to add a 1. so half of 2 is 1 and i’m going to
square that 1 so 1 squared is 1. i’m going to do the same thing for these
two right here so i’m going to take half of the number in front of the y

00:08
which is 14 and i’m going to square that 14 and i’m going to add that
so if you square 14 you’re going to get 196. okay so and then minus 181.
now i added a 1 to this to this part right here
and i added a 196 so i have to do the same over here
now the reason why i did that is because these two
now turn into three and these two turn into three and we can factor
so this would be x minus one squared and this will be x plus fourteen squared
excuse me and then over here we’re going to get sixteen
just by combining those numbers together right there

00:09
so the center is going to be at 1 minus 14 right because you need a negative in
here so we have to make two negatives from that positive and the radius is four
and so now we can go and sketch the graph
so we’re going to be looking at say one and then minus -14 and
so we got the center right there now i just need to make a circle around
that center there and let’s say there’s this there’s the circle right there
all right so it’s kind of a lumpy circle but let’s label that as -14 there
all right so um we completed the square right there and
we completed the square right there that’s a handy trick there called

00:10
completing the square all right so let’s look at a number two here
so number two is similar i’m going to take the same approach
so i’m going to pull this minus 2x with the x squared
and i’m going to leave the 24y with the y squared
and i’m going to move the 120 over to the other side so now i get negative
now i’m going to look at these two and again i’m going to take half of this and
square it so that’s where i get the one from
and in this case i’m looking at half of the 24 which is 12
and i’m going to square that so i get 144. so i added a 1 and 120 and 144
so i’m going to add to the right side also so these two these three right here

00:11
factor x minus 1 squared and these right here factor
and then this right here comes out to be 25.
now if it turns out that it’s negative over here
um then you won’t have a circle right but you know we have positive 25
right so what is the center center is one minus 12 and the radius is five
and so we can go and sketch the graph again
this time i’m going to draw my circle first
not that it makes much of a difference all right so where’s my circle at and
put it right about here there’s the center and that’s at 1
and this is at minus 12 right here there’s my circle right there

00:12
all right so let’s try number three so number three anything going to be
different here let’s check it out so x squared and then we have a minus 28x
and then a y squared and then a minus 10y and then that’s equal to minus 220.
so here i’m going to be looking at half of that 28
and i’m going to be squaring that so 14 so i’m going to be adding a 196.
and here i’m going to be taking half of that 10 and squaring it so i’m going to
be adding a 25 so minus 220 in fact let me just add it all together
we’re getting um minus 220 but i also have to add a 96 and i have to add a 2

00:13
to 25 i have to add this to both sides and i have to add this to both sides so
when i calculate all that up i’m going to be getting one get a one just
enough to be positive so the center is at uh oh i didn’t finish it so this will
be x um minus 14 squared plus y minus five squared equals one so the center is
um fourteen five fourteen five and the radius is um one square root of 1.
all right so we can go and sketch that graph now so we’re way out over here

00:14
14 and then we’re right about here at a 5. so there’s our circle over there
at 14 5 radius is one all right and so um we did several in a row now let’s see
if we can maybe actually try to find something interesting or maybe not let’s
just see what it looks like let’s look at number four here and let’s
just say what if we can throw in some constants here
let’s look at that real quick so we’ll have x squared and then we’ll
pull the 8 a a times x with the x squared and then the y squared and then we’ll
write the b b y with that and then we’re going to move the c to the other side
so how do we uh complete the square here
on these two we’re going to take half of that and square it
so we’re going to be looking at half of a which is a over two sorry you can’t

00:15
see that half of a and square that so that’s a squared over four so i’m
adding a squared over four and i’m going to do the same thing here
this will be b y so i’m going to take half of my b and square it
so this will be plus b squared over four and so i need to add that to both sides
so this will be minus c plus and then we added this so a squared over four
plus b squared over four and so now we can factor these three here
and now we can factor these three here and so we’re going to get x plus
sorry x plus a over 2 squared plus y plus b over 2 squared and then well

00:16
i mean we can write it with the 4 if we want um i’ll just leave all of that
all right so we don’t even know if we have a circle or not so we have circle
circle whenever minus c plus a squared over four plus b squared over four
is greater than zero we have to have that greater than zero here because you
know you can’t have a sum of two squares being equal to a negative number you
just won’t get a radius you won’t get a circle so you have a circle whenever
this is positive and so you can look at the a b and c and you can compute this
number up here and see if it’s positive or not see if you have a circle or not
okay so um that’s it for this video here
i want to say thank you for watching and i’ll see you next time

00:17
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About The Author
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David A. Smith (Dave)

Mathematics Educator

David A. Smith is the CEO and founder of Dave4Math. His background is in mathematics (B.S. & M.S. in Mathematics), computer science, and undergraduate teaching (15+ years). With extensive experience in higher education and a passion for learning, his professional and academic careers revolve around advancing knowledge for himself and others. His work helps others learn about subjects that can help them in their personal and professional lives.

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