Thread: Put a Conic Section into General Form

1. Put a Conic Section into General Form

I have to put the ellipse with the equation $\displaystyle ax^2+by^2+cx+dy+e=0$ into standard form, $\displaystyle \frac{(x-h)^2}{a}+\frac{(y-k)^2}{b}=1$

I have attached my work as a picture since it would take a long time to type in (I did it on Microsoft Office). Could anyone tell me if I am correct, and if not, where did I go wrong? Thanks!

2. Originally Posted by sleigh
I have to put the ellipse with the equation $\displaystyle ax^2+by^2+cx+dy+e=0$ into standard form, $\displaystyle \frac{(x-h)^2}{a}+\frac{(y-k)^2}{b}=1$

I have attached my work as a picture since it would take a long time to type in (I did it on Microsoft Office). Could anyone tell me if I am correct, and if not, where did I go wrong? Thanks!
Hi sleigh,

After completing the square in the 4th row, you factored incorrectly in the 5th row.

It should be:

$\displaystyle a\left(x+\frac{c}{2a}\right)^2+b\left(y+\frac{d}{2 b}\right)^2=-e+\frac{c^2}{4a}+\frac{d^2}{4b}$

Too bad the error came so early. Why on Earth would anyone want to do this, anyway?

3. Originally Posted by masters
Hi sleigh,

After completing the square in the 4th row, you factored incorrectly in the 5th row.

It should be:

$\displaystyle a\left(x+\frac{c}{2a}\right)^2+b\left(y+\frac{d}{2 b}\right)^2=-e+\frac{c^2}{4a}+\frac{d^2}{4b}$

Too bad the error came so early. Why on Earth would anyone want to do this, anyway?
Does this error make the entire thing wrong? Even though I factored wrong, I don't think this affects the rest of the problem because I added the correct amount to the other side, and I do not expand the binomials after I factored them.

I am doing this to program my calculator to put it into standard form and give all of the information. I have a program to do it in standard form, but I wanted it to work for general form, too. This way, for example, the x coorodinate of the vertex is $\displaystyle \frac{-c}{2a}$. This would work for the y coorodinate, $\displaystyle a^2$ term and $\displaystyle b^2$. I already did this for the parabola and circle, but I was unsure of this one. Thanks for your help!