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# Conic section from expanded equation: circle & parabola

Sal manipulates the equation x^2+y^2-3x+4y=4 in order to find that it represents a circle, and the equation 2x^2+y+12x+16=0 in order to find it represents a parabola. Created by Sal Khan.

Video transcript

Let's see if we could do a
couple more of these conic section identification
problems. So I have this problem, x
squared plus y squared minus 2x plus 4y is equal to 4. And so, the first thing I like
to do is just try to figure out what type of conic
section this will be. And we have our -- this
is my x squared term, my y squared term. They're on the same
side of the equation. And they both have
positive coefficients. So this tells me that
we're going to be dealing with an ellipse. And in this case in particular,
the coefficients are the same number. They're both positive 1. So this is going to tell
me that this is a circle. So let's get this in
the standard form. And try to graph the circle. So we're going to want
to complete the square. So let's take the x terms,
since we get x squared minus 2x plus something to complete
the square later on. Plus, and now let's do
the y squared terms. y squared plus 4y plus
something is equal to 4. And now what are we at here? We take 1/2 of the
minus 2, minus 1. Square it. That becomes plus 1. Add a 1. We have nothing out here, so we
really just added a 1 to the left-hand side of
this equation. So we have to add 1 to
the right-hand side. And here we take 1/2 of 4. 1/2 of 4 is 2. 2 squared is 4. Put a 4 here. So you have to add a 4 to the
right-hand side as well. And we actually did add just a
4, because there was nothing multiplying the 4 out here. And so this becomes x minus 1
squared plus y plus 2 squared is equal to 4 plus 1
plus 4 is equal to 9. And there you have it. We have it in the standard
form of a circle. You remember that if a circle
is centered at 0, the standard form would be x squared plus y
squared is equal to r squared. So this is r squared, this
is the radius squared. So that tells us the radius
of the circle is 3. And it was just shifted so that
its origin, instead of being 0, 0, is at the point 1, minus 2. And the reason why we got 1,
minus 2, we just have to think about what makes this
whole expression equal 0? In this case it was the origin. In this case, it's x equals 1. And what makes this whole
expression equal 0, in this case, it was y is equal to 0. In this case it's y
is equal to minus 2. So that's our center. That's our radius, and we're
ready to graph this circle. So it's at, let me see. I should graph the
circle first. That's fair enough. So it's going to
be at 1, minus 2. So, 1, minus 2. So it's going to be down here. So it's going to come out, this
circle's going to start there. It's going to be at 1,
and then you go 1, 2. So that's pretty
close to the center. Maybe I should do it at 1, 2. So that's at center. Right there, at 1, minus 2. And then its radius is 3. So this distance right here
is 3, in any direction. 3. And that is 3. Fair enough, that was a pretty
straightforward problem. Circles in some ways
are the simplest. And remember I said it's
going to be an ellipse. And you say, oh, this isn't
the standard formula for an ellipse. So just as a refresher, if you
divide both sides of this equation by 9, what do you get? You get x minus 1 squared
over 9 plus y plus 2 squared over 9 is equal to 1. And then you see that the
horizontal axis, I guess. Or the horizontal diameter,
is going to be 3. Or the horizontal radius
is going to be 3. And the vertical radius
is also going to be 3. Because the radius never
changes in this ellipse. Which is really a circle. Let's do one more. Just to make sure you
know this stuff cold. So I have 2x squared plus y
plus 12x plus 16 is equal to 0. Let's look at the x squared
and the y squared terms. 0's an squared term. But I don't see a
y squared term. So this is a bit
of a conundrum. And this will lead us to the
fourth of our conic sections, which I talk about in the
first video but we haven't really touched on yet. And that's the parabola. And how do I know
it's a parabola? You're familiar, and I'll go
more in future videos on all the different ways that
a parabola comes about. And how all the points are
equidistant between one point and a line and all of that. But just in very simple ways,
you recognize the most simple parabola is y
is equal to x squared. That parabola looks
something like this. Where its minimum point, or
its vertex, is at the origin. Or if you have a parabola like,
x is equal to y squared. That looks something like this. Where it's a sideways
version of that one. Where, once again, its
vertex is at the origin. And just out of interest, so we
know that this is a parabola, because we have a y and we
have an x squared, right? They're different degrees. There's no second
degree term of the y. And just to put this in a form
that's familiar to you, let's just subtract everything but
the y from the left-hand side. So you get y is equal to minus
2x squared minus 12x minus 16. And this is kind of the
traditional form that you're familiar with. You're probably even used to
finding the 0'z of this parabola, and we could
do that right now. We could say, OK, when
does this equation intersect the x axis. The x axis is when
y is equal to 0. So that equals 0. You get minus 2x squared
minus 12x minus 16. And this is different to
what we normally do. Normally I would immediately
break into completing the square. But I just want to figure out
the 0's of this parabola first. So this 0's equal to minus 2
times factoring out minus 2, you get x plus 6x plus 8. So the 0 is equal to minus 2
times x plus 2 times x plus 4, I just factored that. And so for this whole thing
to be 0, either this is 0 or that is 0. And so, either x plus
2 is equal to 0, or x plus 4 is equal to 0. So x is equal to minus 2,
and x is equal to minus 4. That's the two 0's
of this parabola. So we immediately know one
thing about this parabola, and you've probably already done
this in your algebra classes. If we were to draw the x axis,
it intersects the x axis at 1 and minus 2, and 3 and minus 4. That's all we know
about this right now. So let's see if we can use some
of our completing the square skills with the conic sections
we've done so far, to to come up with a little bit more
information about this parabola. So let's try to complete
the square with it. I'll rewrite it down here. So it's y is equal to, this
is the one I'm dealing with. Now, let me just take the x
terms by themselves, and factor out the minus 2. Minus 2 times x
squared plus 6x. And I'm going to add
something else. And then I have a
minus 16 over there. To make this a perfect square,
I have to take 1/2 of this, 1/2 of the 6. 1/2 of the 6 is 3. 3 squared is 9. If I add nine to the right-hand
side of the equation, remember, I didn't just add 9. This is 9 times
minus 2 I'm adding. So if I subtract, that's minus
18, if I subtract 18 from the right-hand side, I also have to
do it from the left-hand side. So, subtract 18 there. And now my equation becomes y
minus 18 is equal to minus 2 times, what is this? This is x plus 3
squared minus 16. And let's just get it in a form
that we might start recognizing from our conic sections. Let's add 16 to both sides. If we add 16 to both sides,
y minus 18, plus 16. It's going to be y minus
2, I'll put parentheses around that. Is equal to minus 2
times x plus 3 squared. And you might wonder why
I put it in this form. And I did because this'll help
us, this is kind of the same pattern that you see with all
of the other conic sections. Like, if I were to tell you to
graph y is equal to x squared, y is equal to x squared. It would look something like,
let me draw the - whoops - Draw some axes here. y equals x squared looks
something like this. It looks like a parabola. I mean, it is a parabola,
with the vertex at 0, or its minimum point. And that's what the vertex is,
is the minimum point of, or the maximum point of the parabola. We'll talk more about that,
you'll learn a lot more about that when we go into calculus. But I think you can recognize
it, the bottom of the U or the top of the U. If I were to draw y is equal
to minus x squared, you could plot some points. But it looks
something like that. If I were to try to graph y is
equal to 2x squared, it would just be like the y is equal to
x squared, but it would go up twice as fast. It would look
something like that. So the vertex at the origin. And finally, if I were to
graph y is equal to minus 2x squared, it would look
something like this. It would open downward and it
would go down twice as fast. Now, this one, this equation
that we ended up with, right here, this is the same thing as
y is equal to minus 2x squared. It has the same general shape. But instead of its vertex, or
its center or its starting point, or whatever you want
to call it, to be at the origin, it's shifted now. You say what y value makes
this term zero, well, y is equal to 2. Just why, in this case,
what y value makes this 0? Well, it's 0 here because
we were at the origin. And here, what x
value makes this 0? It's x is equal to minus 3. So this gives us information
about where the vertex is. It's at x is equal to
minus 3, y is equal to 2. So it's at x is equal to 1,
2, 3. y is equal to 1, 2. Right there. We already know those two
points, because we've figure out it's 0. But even if we didn't know
those two points -- we know that this has the same
general shape as y is equal to minus 2x squared. So it's going to open downward. Like this. And a little bit faster than y
is equal to minus x squared. So it's going to
look like this. And we know it goes through
that point and we know it goes through that point. There you go. So we've touched on
every conic section. And in the next two videos,
I'll go into a little bit more depth of the theory behind
conic sections and how they came about and all of that. But I think now you're ready
to tackle a lot of what you might actually see on
your algebra test. See you soon.