how to make equations of conics easier to read if

Question

How to Make Equations of Conics Easier to Read ?

If you want to graph a conic sections, first you need to make the equation easy to read. For example, say you have the equation

4x² + 64 = 40x + 9y² .
You know that it's a conic section, because it's second-degree- in fact, if you've read the rest of this chapter, you can even figure out that it's a hyperbola. But it's not obvious what the graph looks like. Where's the center for example?

To make this equation easier to read, we need to get rid of the first-degree terms, by completing the square!

For example, the term - 40 x is a first-degree term. Let's get rid of it. Move it over to the left with the other x term (while we're at it, we'll go ahead and move all the variable to the left):
4x² - 40x - 9y² = -64.
Important: factor out the coefficient of x2 (from the x terms only) before attempting to complete the square!

4(x² -10x) - 9y² =-64.
Now, complete the square inside the parentheses.

4(x² - 10x)-9y² = -64
4((x-5) 2 - 25) - 9y² = -64

and re-distribute the 4 (pay special attention to the way the 4 gets distributed to the new constant, -25, created when we completed the square!).

4(x - 5) 2 - 100 - 9y² = -64
Then combine the constant terms:

4(x - 5) 2 - 9y² = 36.
For this equations, there's no need to complete the square for the y terms, because there is no first-degree y term. Now divide both sides by the constant term:

(x- 5)2/9 -y² /4 = 1

and finally, take the square roots of the constant factors 9 and 4 to bring them under the square symbol:

(x-5/3)2 -(y/2)2 = 1 (1)

I know, I know! You're saying, "what in the world is all this for?!" Well, take a look at the result (equation 1). It's in a very simple form. There are only three terms. One of the terms has only the variable x and is squared; same for the variable y. The constant term is just 1. In fact, the equation has been made as close as possible to the equation of the "standard" hyperbola,

x² - y² = 1,

except for some translation and scaling factors. (The graph is translated in the x-direction a distance 5, and is scaled in the x and y directions by factors of 3 and 2 respectively. I'm deliberately not showing you the graph here, because I want you to look at the equations!)