# conic standard form/conversion

• Feb 17th 2010, 08:56 AM
koumori
conic standard form/conversion
I have a few conic questions that I need some help with as they are not in a format that I understand to be the normal for conics.
I already know what type of conic each one stands for because I used Microsoft math to draw them, but I would like to solve them for myself.

1
$\frac{(x-3)^2}{9}+\frac{(y-2)^2}{4}=1$
it stands for an ellipse

2
$\frac{(x+2)^2}{25}-\frac{(y-1)^2}{9}=1$
It stands for a hyperbola

3
$x^2+4x=y-4$
It stands for a parabola
• Feb 17th 2010, 09:10 AM
HallsofIvy
I'm going to assume that there is no "xy" term in your equations- that would be a "tilted" conic (axes NOT parallel to the x and y axes) and that is much more complicated.

Anything that has either x or y squared but not both is a parabola.

Anything that has both x and y squared and the coefficients of the squares, on the same side of the equal sign, of opposite sign, one positive and one negative, is a hyperbola.

Anything that has both x and y squared and the coefficients of the squares, on the same side of the equal sign, of the same sign, both positive or both negative, is either a circle or an ellipse. If the coefficients are the same number, then it is a circle. If the coefficients have the same sign but different values, then it is an ellipse. (Some people consider the circle a special case of an ellipse.)
• Feb 17th 2010, 12:52 PM
koumori
OK! Thanks!

My big problem is that while I was taught how to work out a problem dealing with conics, I have no idea what to do with these equations, I know I can't just solve for x--but I am having some trouble working these questions into a format I know what to do with.
I was asked to show all working, so is there a way to show how one knows what type of conic it is from these particular equations?
• Feb 17th 2010, 01:33 PM
Diagonal
Quote:

Originally Posted by koumori
OK! Thanks!

My big problem is that while I was taught how to work out a problem dealing with conics, I have no idea what to do with these equations, I know I can't just solve for x--but I am having some trouble working these questions into a format I know what to do with.
I was asked to show all working, so is there a way to show how one knows what type of conic it is from these particular equations?

What you need to do is to study "families" of curves to understand what each parameter does as it changes. You would then see, for example, that

$

\frac{(x-3)^2}{9}+\frac{(y-2)^2}{4}=1
$

is the same curve as $

\frac{x^2}{9}+\frac{y^2}{4}=1
$

but shifted [translated] by 3 in the x-direction and 4 in the y-direction. As I suggest, this deserves a study of several curves watching their behaviour. The same principles apply to other curves [equations] as they relate to their "standard" form. You might start off with the line and the circle as well to observe what happens with translation and rotation.

I hope this helps.