Identifying Conics 1 Part 1 of identifying and graphic conic sections
Identifying Conics 1
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- The standard question you often get in your algebra class is
- they will give you this equation and it'll say identify
- the conic section and graph it if you can.
- And the equation they give you won't be in the standard form,
- because if it was you could just kind of pattern match with
- what I showed in some of the previous videos and you'd
- be able to get it.
- So let's do a question like and let's see if
- we can figure it out.
- So what I have here is 9x squared plus 4y squared
- plus 54x minus 8y plus 49 is equal to 0.
- And once again, I mean who knows what this is it's just
- not in the standard form.
- And actually one quick clue to tell you what this is you look
- at the x squared and the y squared terms if there are.
- If there's only an x squared term and then there's just a y
- and not a y squared term, then you're probably dealing with a
- parabola, and we'll go into that more later.
- Or if it's the other way around, if it's just an x
- term and a y squared term, it's probably a parabola.
- But assuming that we're dealing with a circle, an ellipse, or a
- hyperbola, there will be an x squared term and a
- y squared term.
- If they both kind of have the same number in front of them,
- that's a pretty good clue that we're going to be
- dealing with a circle.
- If they both have different numbers, but they're both
- positive in front of them, that's a pretty good clue
- we're probably going to be dealing with an ellipse.
- If one of them has a negative number in front of them and
- the other one has a positive number, that tells you that
- we're probably going to be dealing with a hyperbola.
- But with that said, I mean that might help you identify things
- very quickly at this level, but it doesn't help you graph it or
- get into the standard form.
- So let's get it in the standard form.
- And the key to getting it in the standard form is really
- just completing the square.
- And I encourage you to re-watch the completing the square
- video, because that's all we're going to do right here to get
- it into the standard form.
- So the first thing I like to do to complete the square, and
- you're going to have to do it for the x variables and for
- the y terms, is group the x and y terms.
- Let's see.
- The x terms are 9x squared plus 54x.
- And let's do the y terms in magenta.
- So then you have plus 4y squared minus 8y and then you
- have-- let me do this in a different color-- plus
- 49 is equal to 0.
- And so the easy thing to do when you complete the square,
- the thing I like to do is, it's very clear we can factor out a
- 9 out of both of these numbers, and we can factor out a
- 4 out of both of those.
- Let's do that, because that will help us
- complete the square.
- So this is the same thing is 9 times x squared plus
- 9 times 6 is 54, 6x.
- I'm going to add something else here, but I'll
- leave it blank for now.
- Plus 4 times y squared minus 2y I'm probably going to add
- something here too, so I'll leave it blank for now.
- Plus 49 is equal to 0.
- So what are we going to add here?
- We're going to complete the square.
- We want to add some number here so that this whole three term
- expression becomes a perfect square.
- Likewise, we're going to add some number here, so this
- three term number expression becomes a perfect square.
- And of course whatever we add on the side, we're going
- to have to multiply it by 9, because we're really
- adding nine times that.
- And add it on to that side.
- Whatever we add here, we're going to have to multiply
- it times 4 and add it on that side.
- If I put a 1 here, it's really like as if I had a 4 here,
- because 1 times 4 is 4 and if I had a 1 here it's 1 times 9.
- So 9 there.
- Let's do that.
- When we complete the square, we just take half of
- this coefficient.
- This coefficient is 6, we take half of it is 3, we
- square it, we get a 9.
- Remember it's an equation, so what you do to one side, you
- have to do to the other.
- So if we added a 9 here, we're actually adding 9 times 9 to
- the left-hand side of the equation, so we have to add 81
- to the right-hand side to make the equation still hold.
- And you could kind of view it if we go back up here.
- This is the same thing, just to make that clear as if I
- added plus 81 right here.
- Of course I would have had to add plus 81 up here.
- Now let's go to the y terms.
- You take half of this coefficient is minus 2,
- half of that is minus 1.
- You square it, you get plus 1.
- 1 times 4, so we're really adding 4 to the left-hand
- side of the equation.
- And just so you understand what I did here.
- This is equivalent as if I just added a 4 here, and then I
- later just factored out this 4.
- And so what does this become?
- This expression is 9 times what?
- This is the square of-- you could factor this, but we did
- it on purpose-- it's x plus 3 squared and then we have plus
- 4 times-- What is this right here?
- That's y minus 1 squared.
- You might want to review factoring of polynomial or
- completing the square if you found that step
- a little daunting.
- And then we have plus 49 is equal to 0 plus 81 plus
- 84 is equal to 85.
- All right, so now we have 9 times plus 3 squared plus 4
- times y minus 1 squared.
- And let's subtract 49 from both sides.
- That is equal to-- let's see if I subtract 50 from 85 I get 35,
- so if I subtract 49, I get 36.
- And now we are getting close to the standard form of something,
- but remember all the standard forms we did except for the
- circle-- we had a y-- and we know this isn't a circle,
- because we have these weird coefficients, well not weird
- but different coefficients in front of these terms.
- So to get the 1 on the right-hand side let's
- divide everything by 36.
- If you divide everything by 36, this term becomes x plus 3
- squared over see 9 over 36 is the same thing as 1 over 4, and
- then you have plus y minus 1 squared 4 over 36 is the same
- thing as 1 over 9 and all of that is equal to 1.
- And there you go.
- We have it in the standard form, and you can see our
- intuition at the beginning the problem was correct.
- This is indeed an ellipse, and now we can actually graph it.
- So first of all, actually good place to start, where's
- the center of the this ellipse going to be?
- It's going to be x is equal to negative 3.
- What x value makes this whole terms 0?
- So it's going to be x is equal to minus 3, and y is
- going to be equal to 1.
- What y value makes this term 0? y is equal to 1.
- That's our center.
- So let's graph that, and then we can draw the ellipse.
- It's going to be in the negative quadrant.
- This is our x-axis and this is our y-axis.
- And then the center of our ellipse is at minus 3
- and positive 1, so that's the center.
- And then, what is the radius in the x direction?
- We just take the square root of this, so it's 2.
- So in the x direction we go two to the right.
- We go two to the left.
- And in the y direction, what do we do?
- Well we go up three and down three.
- The square root of this.
- Let me do that.
- Remember you have to take the square root of both of those.
- The vertical axis is actually the major radius or the
- semi-major axis is 3, because that's the longer one.
- And then the 2 is the minor radius, because that's
- the shorter one.
- And now we're ready to draw this ellipse.
- I'll draw it in brown.
- Let me see if I can do this properly.
- I have a shaky hand.
- All right, it looks something like that.
- And there you go.
- We took this kind of crazy looking thing, and all we did
- is algebraically manipulate it.
- We just completed the squares with the x's and the y terms.
- And then we divided both sides by this number right here and
- we got it into the standard form.
- We said oh this is an ellipse.
- We have both of these terms, they're both positive, we're
- adding we're not subtracting, they have different
- coefficients underneath here.
- So we're ready to go over the ellipse, and we realized that
- the center was at minus 3,1, and then we just drew the
- major radius, or the major axis and the minor axis.
- See you in the next video.
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