1. ## rotation of axes

Given the equation 21x^2 + 10 root3xy + 31y^2 = 144

a. determine the type of conic using the discriminant
b.find the angle of rotation
c.use rotation formulas to eliminate the xy-term and sketch

I was assigned a bunch of problems just like this for homework. The only problem is that my book doesn't really give an example of this type of problem, nor did my teacher in class. If anyone could help my figure this out I would be sooo grateful. Thanks so much!

2. Originally Posted by kristenrae
Given the equation $\displaystyle 21x^2 + 10\sqrt{3}xy + 31y^2 = 144$

a. determine the type of conic using the discriminant
b.find the angle of rotation
c.use rotation formulas to eliminate the xy-term and sketch
a. determine the type of conic using the discriminant

The general equation for conic sections can be written as
$\displaystyle Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$

Comparing the given equation with the general equation,
$\displaystyle A = 21, B = 10\sqrt{3},\ C = 31, D = 0, E = 0,\ \mbox{and}\ F = -144$ ------- ■

The discriminant is
$\displaystyle B^2 - 4AC = (100\cdot 3) - (4\cdot 21\cdot\ 31) < 0$

$\displaystyle B^2 - 4AC < 0.$ Therefore, the equation represents an $\displaystyle \boxed{\mbox{ellipse}}$.

b.find the angle of rotation

For the general equation of a conic, the angle of rotation $\displaystyle \theta$ about the origin is given by:

$\displaystyle \theta = \frac{\pi}{4}$, if A = C
$\displaystyle \tan 2\theta = \frac{B}{A - C}$, if $\displaystyle A\neq C$

In this case, $\displaystyle A\neq C$
Substituting 21 for A, $\displaystyle 10\sqrt{3}$ for B, and 31 for C,

$\displaystyle \tan 2\theta = \frac{10\sqrt{3}}{21 - 31} = -\sqrt{3}$

Solving for $\displaystyle \theta$,

$\displaystyle 2\theta = \arctan (-\sqrt{3}) = -60^\circ$
$\displaystyle \implies \boxed{\theta = -30^\circ}$ -------●

c.use rotation formulas to eliminate the xy-term and sketch

If the conic section with the equation:$\displaystyle Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$

is transformed by a rotation about the origin through the angle $\displaystyle \theta$, then the equation in the new coordinates $\displaystyle \tilde{x}$ and $\displaystyle \tilde{y}$ has the form:

$\displaystyle \tilde{A}\tilde{x^2} + \tilde{C}\tilde{y^2} + \tilde{D}\tilde{x} + \tilde{E}\tilde{y} + \tilde{F} = 0$

The coordinate transformation which expresses the old coordinates in terms of the new ones is given by:

$\displaystyle x = \tilde{x}\cos \theta - \tilde{y}\sin \theta$
$\displaystyle y = \tilde{x}\sin \theta + \tilde{y}\cos \theta$

Thus, putting the coordinate transformations into the original equation:

$\displaystyle Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$

$\displaystyle \Rightarrow A(\tilde{x}\cos \theta - \tilde{y}\sin \theta)^2 + B(\tilde{x}\cos \theta - \tilde{y}\sin \theta)(\tilde{x}\sin \theta + \tilde{y}\cos \theta)$ $\displaystyle +\ C(\tilde{x}\sin \theta + \tilde{y}\cos \theta)^2 + D(\tilde{x}\cos \theta - \tilde{y}\sin \theta) + E(\tilde{x}\sin \theta + \tilde{y}\cos \theta) + F = 0$

Opening up the brackets and expanding, we get:

$\displaystyle A\tilde{x}^2\cos^2 \theta + A\tilde{y}^2\sin^2 \theta - 2A\tilde{x}\tilde{y}\cos \theta\sin \theta + B\tilde{x}^2\sin \theta \cos \theta + B\tilde{x}\tilde{y}\cos^2 \theta$ $\displaystyle \ - B\tilde{x}\tilde{y}\sin^2 \theta - B\tilde{y}^2\sin \theta\cos \theta + C\tilde{x}^2\sin^2 \theta + C\tilde{y}^2\cos \theta$

$\displaystyle +\ 2C\tilde{x}\tilde{y}\sin \theta\cos \theta + D\tilde{x}\cos \theta - D\tilde{y}\sin \theta + E\tilde{x}\sin \theta + E\tilde{y}\cos \theta + F = 0$

Rearranging,
$\displaystyle \tilde{x}^2(A\cos^2\theta + B\sin \theta\cos \theta + C\sin^2 \theta) + \tilde{y}^2(A\sin^2\theta - B\sin \theta\cos \theta + C\cos^2 \theta)$ $\displaystyle +\ \tilde{x}\tilde{y}\{\sin \theta\cos \theta(2C - 2A) + B(\cos^2 \theta - \sin^2 \theta)\} + \tilde{x}(D\cos \theta + E\sin \theta)$ $\displaystyle +\ \tilde{y}(E\cos \theta - D\sin \theta) = 0$

Thus, the parameters of the new equation are:

$\displaystyle \tilde{A} = A\cos^2 \theta + B\cos \theta\sin \theta + C\sin^2 \theta$

$\displaystyle \tilde{B} = 0$ $\displaystyle [\mbox{Using the identities}\ \sin 2\theta = 2\sin \theta\cos \theta\ \mbox{and}\ \cos 2\theta = \cos^2 \theta - \sin^2 \theta$ and using ■ and ●, we get $\displaystyle (C - A)\sin 2\theta + B(\cos 2\theta) = -\frac{\sqrt{3}}{2}\times 10 + 10\sqrt{3}\times \frac{1}{2} = 0]$

$\displaystyle \tilde{C} = A\sin^2 \theta - B\cos \theta\sin \theta + C\cos^2 \theta$

$\displaystyle \tilde{D} = D\cos \theta + E\sin \theta$

$\displaystyle \tilde{E} = E\cos \theta - D\sin \theta$

$\displaystyle \tilde{F} = F$

In our case, applying the rotation formula,

$\displaystyle \boxed{x = \frac{\sqrt{3}}{2}\tilde{x} + \frac{1}{2}\tilde{y}}$
$\displaystyle \boxed{y = -\frac{1}{2}\tilde{x} + \frac{\sqrt{3}}{2}\tilde{y}}$

$\displaystyle \tilde{A} = \frac{63}{4} - \frac{15}{2} + \frac{31}{4} = 16$

$\displaystyle \tilde{C} = \frac{21}{4} + \frac{93}{4} + \frac{30}{4} = 36$

$\displaystyle \tilde{D} = 0$

$\displaystyle \tilde{E} = 0$

$\displaystyle \tilde{F} = -144$

and the equation gets the form:

$\displaystyle 16\tilde{x^2} + 36\tilde{y^2} - 144 = 0$

$\displaystyle \implies \boxed{\frac{\tilde{x}^2}{9} + \frac{\tilde{y}^2}{4} = 1}$,

which is the equation of a standard ellipse.

3. Thank you so much! That was so helpful. I understand completely up until applying the rotation formula. How did you find the values of A through F? Thanks so much again!