Graphing a Rotated Conic on a Graphing Calculator

1. Jul 18, 2010

1. The problem statement, all variables and given/known data
Use a graphing utility to graph the conic. Determine the angle through which the axis are rotated.

$$x^2+xy+y^2=10$$

2. Relevant equations
$$\cot2\theta=\frac{A-C}{B}$$
$$x=x'\cos\theta-y'\sin\theta$$
$$y=x'\sin\theta+y'\cos\theta$$

3. The attempt at a solution

I can find the angle of rotation to be 45 degrees and I know the equation in terms of x' y' is.

$$\frac{(x')^2}{\frac{20}{3}}+\frac{(y')^2}{20}=1$$

However I don't know how to graph the rotated function. My graphing calculator can not graph implicit functions. I can get function in terms of x',y' in terms of y' and graph that, but that wont be on an x',y' axis on my calculator, only an x,y axis. How can I graph the rotated function on a calculator?

2. Jul 18, 2010

eumyang

To graph the conic using a graphing utility, you need to solve the conic for y. Rewrite the equation as a quadratic equation to y (the x's will be part of the coefficients):
\begin{aligned} x^2 + xy + y^2 &= 10 \\ y^2 + (x)y + (x^2 - 10) &= 0 \\ \end{aligned}
Use the quadratic formula with $$a = 1$$, $$b = x$$, and $$c = (x^2 - 10)$$.

69

3. Jul 18, 2010

I did that and I wind up getting an imaginary answer. My calculator comes up with errors.

$$x=\frac{-x\pm\sqrt{-3x^2-40}}{2}$$

4. Jul 18, 2010

eumyang

(I assume you meant to type "y" on the left side.) Should be + 40, not - 40. Underneath the square root you should have set up
$$\sqrt{x^2 - 4(1)(x^2 - 10)}$$

69

Last edited: Jul 18, 2010
5. Jul 18, 2010