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#2

# Example to show y = 2x^2 + nx
+ 3x - 1

is a hyperbola for a specific value of
n.

I will use the standard way given in many
precalculus and calculus books to find a rotation of the axes
so that the xy term disappears. In other words, I will follow
the standard way to "get rid of the xy term" in a conic
relation by trying to find x' and y' axes that are rotated with
respect to the x and y axes.*

In particular, to proceed, consider the
conic relation Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0.

Let **x = x'cos****q - y'sinq**

and let **y = x'sin****q + y'cosq.**

Also let tan2q = B/(A - C). (This equation is a standard way to
find the rotation angle, q. For more details, see the reference below or almost
any calculus or honors precalculus textbook.)

With our equation, **2x^2 + nxy + 3x - y - 1 = 0**, C is always
0; A = 2; D = 3; E and F both equal -1; and B = **n**.

Therefore, tan2q = B/A = **n**/2.

To view an example of how this process works,
let B = **n**
= 2sqrt(3). Then tan2q = 2sqrt(3)/2 = sqrt(3) and so 2q = p/3.
Therefore, q
= p/6; cosq = sqrt(3)/2; and sinq = 1/2.

Now, that means we will replace **x**
with **x'sqrt(3)/2 - y'/2**

and **y** with
**x'/2 + y'sqrt(3)/2**.

The result is:

2(x'sqrt(3)/2 - y'/2)^2 + **2sqrt(3)**(x'sqrt(3)/2
- y'/2)(x'/2 + y'sqrt(3)/2) + 3(x'sqrt(3)/2 + y'/2) - (x'/2 +
y'sqrt(3)/2) - 1 = 0

3x'^2/2 - sqrt(3)x'y' + y'^2/2 + **2sqrt(3)**(sqrt(3)x'^2/4
+ x'y'/2 - sqrt(3)y'^2/4) + 3sqrt(3)x'/2 + 3y'/2 - x'/2 - y'sqrt(3)/2
- 1 = 0

3x'^2/2 **- sqrt(3)x'y'** + y'^2/2
+ 3x'^2/2
**+ sqrt(3)x'y'** - 3y'^2/2 + 3sqrt(3)x'/2 + 3y'/2 - 1 = 0

Notice that the x'y' terms will disappear,
leaving

**3**x'^2 **-
1**y'^2 + (3sqrt(3)/2 - 1/2)x' + (3/2
- sqrt(3)/2)y' - 1 = 0

And indeed, this equation is a hyperbola
because the coefficients of x^2 and y^2 are both non-zero and
are opposite in sign. (Of course, this sort of criterion can only
be used when there is no xy term!)

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*I referenced the following book in this analysis:

Stein, S. K. (1987). __Calculus and Analytic Geometry__.
New York: McGraw-Hill Book Company.