Quadadic Relations and Systems - Conic sections - Resolved

[Xearf_987]

In my Algebra 2 class, we're studying conic sections. I'm not so sure I know how to explain exactly what we're doing, so I'll just give the instructions.

Find a polynomial equation with integer coefficents for the set of coplaner points described. Tell whether or not the graph is a conic section (circle, ellipse, hyperbola, parabola) and, if it is, tell which conic section. The point given in each problem is called the focus and the line given is called the directrix.

This may seem like a silly question, but I'm just asking so that I can be 100% sure of myself.

The problem I'm on says:

Each point is equidistant from the point (2,5) and the line x = -3

I'm suppose to derive an equation from this information which will tell me if it's a conic section or not.


My teacher said that it would help if we drew a coordinate plane and drew in the information. Then, picking any random point, calling it (x,y), that might be on the graph, draw... in this case.... equal length lines from the focus and the directrix
to (x,y). Now, when I set up my distance formuals... I wasn't quite sure about something.

Basically, my question is: should I set up the problem like this

Sqrt( (x+3)^2 + (y-y)^2 ) = Sqrt( (x-2)^2 + (y-5)^2 )

Or should it look like

Sqrt( (x+3)^2 + (y-y)^2 ) = Sqrt( (2-2)^2 + (y-5)^2 )

Or are they both wrong? (I bolded the only difference between the two)

After simplifying the first one, the equation I got was: y^2 - 10x - 10y + 20, which I don't think is a conic section.

And after doing the last one, I got the equation: -x^2 + y^2 - 6x -10y +16, which would be a hyperbola.


I know this sounds a bit confusing. If you can't figure out what I'm trying to say, then please tell me what more I should add so that this makes sense.

 

[pka]

By definition a parabola is a set of points in the plane that are equidistance from a fix point and a fixed line.

 

[soroban]

Re: Quadadic Relations and Systems - Conic sections

Hello, Xearf_987!

Find a polynomial equation with integer coefficents for the set of coplaner points described.
Tell whether or not the graph is a conic section (circle, ellipse, hyperbola, parabola) and, if it is, tell which conic section.
The point given in each problem is called the focus and the line given is called the directrix.

Each point is equidistant from the point (2,5) and the line x = -3

\(\displaystyle \sqrt{(x+3)^2\,+\,(y-y)^2}\;=\;\sqrt{(x-2)^2\,+\,(y-5)^2}\;\) . . . This one is correct

After simplifying, I got: \(\displaystyle y^2\,-\,10x\,-\,10y\,+\,20\) = 0, which I don't think is a conic section. \(\displaystyle \;\) . . . It is!

          :       |
    (-3,y)* - - - + - - - *(x,y)
          :       |      /
          :       |     /
          :       |    /
          :       |   *(2,5)
          :       |
      ----+-------+------------
         -3       |

An conic has an equation in wnich at least one of the variables is squared.

We can "eyeball" the equation if we know some rules . . .


Exactly one variable squared \(\displaystyle (x^2\) or \(\displaystyle y^2)\): parabola.


Both squared \(\displaystyle (x^2\) and \(\displaystyle y^2)\)

\(\displaystyle \;\;x^2\) and \(\displaystyle y^2\) have the same sign: ellipse.

\(\displaystyle \;\;x^2\) and \(\displaystyle y^2\) have the same cofficient: circle.

\(\displaystyle \;\;x^2\) and \(\displaystyle y^2\) have opposite signs: hyperbola.

 

[stapel]

Your first equation is the one you should use.

Eliz.

 

[Xearf_987]

Okay, so then it's a parabola... correct? I guess I'm just not used to seeing both an x and a y term.