Welcome, ladies and gentlemen.
So what I would like to do is show you
how to classify conic sections.
And basically, we're going to classify conic selections
either as a parabola, a circle, an ellipse, or a hyperbola.
And so what I did is I took our general equation
of a conic section, ax squared plus by squared plus cx plus dy
plus e, where a, b, c, d, and e are all
going to be real numbers.
And what we did is I also then wrote in the common equations
that we use for a parabola, a circle, an ellipse,
an hyperbola, to help me as I'm explaining things.
But basically, what the purpose comes
into is, if you have an equation that's in this format,
and what you notice is a times b, if you multiply those,
you're going to get 0.
Now, the reason why this works for a parabola, the reason why
a times b equals 0 is going to tell you
it's a problem, because, when you look at this, if you were
to expand-- we've got to think about all
these binomial squareds we're going to be expanding-- if you
expanded this out, you would only have y squared.
You would not have an x squared.
So if you have a no x squared for that--
if you don't have an x squared, that means
the coefficient has to be 0.
So that means one of these values is 0.
If one of those values is 0, that
means the product of any number times
its value is going to be 0.
In the case of a circle, remember,
if we want to set it equal to 1, if I was going to divide
by r squared, divide on both sides,
you would notice the coefficients of my x
and my y, even once I expand them,
are going to be exactly the same when you have a circle.
So therefore, the coefficients of a and b
are always going to be the same.
That's why when a is equal to b, you have a circle.
For an ellipse, the main thing I look at
is noticing the difference between ellipses,
no matter what expansion of my x and my y's, it's always adding.
That means I'm always going to have
a positive coefficient for x and a positive coefficient for y.
So an ellipse, if you multiply your a times b,
and it's greater than 1, two positives multiplied together
is always going to give you a positive.
Whereas, a hyperbola, one's positive , one's negative.
One's positive and one's negative.
So if your coefficients, one's positive,
one's negative, whatever you multiply
them to get your product, it's always going to be less than 1.
So that's just the way I internalize and think
about these things.
Now, let's get through some problems here.
So the main important thing that you
want to do in classifying is identify your a and your b.
So here, I notice that my a and my b are exactly the same.
So if I was going to complete the square
and do all the stuff, which you could do and try
to write into this form and see which one it works,
that's just a lot of work.
I notice that my a and my b are exactly the same.
Whenever my a and my b are exactly the same,
I know it's going to be a circle,
because, again, divide by r squared, set it equal to 1,
you're going to have 1 over r squared and 1 over r squared.
It's always going to be exactly the same coefficient,
even once you expand it out.
So therefore, that gives you a circle.
In the next example, I have my a here is 1 and b is 4.
They're not the same, so it's not a circle.
They don't multiply.
1 times 4 does not multiply to give me 0.
But 1 times 4 multiplies to give me
a positive number, which is 4, which is
going to make this an ellipse.
Again, if you were to complete the square,
you would have the addition between your x and your y
terms, but they wouldn't be exactly the same.
So that's why that's an ellipse.
The next one, I have a and b.
Obviously, you can see this one, they're not the same.
So it's not the circle.
They don't multiply to give you 0.
And when you multiply them, 4 times negative 2,
you're going to get a number that's less than 1-- oh, that's
greater than 0.
I don't know where 1 came from.
So that's going to give me a hyperbola, because 4
times negative 2 is negative 8.
Negative 8 is less than 0.
And again, if you were to group these and complete the square,
they would have different coefficients,
but one would be positive, one would be negative.
And in the last example, you see here-- is you can say,
all right, well, I have an a, but where is b?
There is no b, right?
There is no x squared.
So therefore, b has to equal 0, right?
It has to be 0x squared.
So therefore, a, which is 1, 1 times 0 is going to give you 0.
So therefore, that tells us that we're going to have a parabola.
So the easiest way to classify is just
to remember each one of these.
But I think the process of actually completing the square
and working through them will also
help you remember how to classify your conic sections.
Thanks.