1. ## Conic Sections

Hi.

I'm just starting to learn about conic sections, and it says in my text that their general equation is $\displaystyle a^2+by^2+2gx+2fy+c=0$.

Is that an error? Isn't it supposed to be $\displaystyle ax^2$ not just $\displaystyle a^2$ ?

2. Yes, the x-squared is missing. Also, the xy term is missing. In my book the general form of a quadratic equation in 2 variables is
$\displaystyle Ax^2 +Bxy +Cy^2 + Dx + Ey + F = 0$.

After a Google Search, I have found this equation also expressed as
$\displaystyle ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$,
which is closer to what you have.

01

3. Awesome. Thanks for that.

Would the $\displaystyle xy$ term be present for hyperbolas and ellipses also? As my text says that this general equation is for all conic sections...

4. Originally Posted by Stroodle
Awesome. Thanks for that.

Would the $\displaystyle xy$ term be present for hyperbolas and ellipses also? As my text says that this general equation is for all conic sections...
Yes.

Actually, now that I think about it, if there is an xy term then the conic is rotated by some angle. You stated that you're starting to learn about conic sections, so the book probably omitted the xy term deliberately. This means that in parabolas & hyperbolas, without the xy term, the curves will only open upward/downward/left/right. And in ellipses, the major/minor axes will only be horizontal/vertical lines.

01

5. Cool. Thanks for your help!