Graphing a Rotated Conic on a Graphing Calculator

[themadhatter1]

Homework Statement

Use a graphing utility to graph the conic. Determine the angle through which the axis are rotated.

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

Homework Equations

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

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 won't be on an x',y' axis on my calculator, only an x,y axis. How can I graph the rotated function on a calculator?

 

[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

 

[themadhatter1]

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

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

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

 

[eumyang]

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

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

(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

 

[themadhatter1]

(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

Thanks for all your help. It works fine now.