# How do you rotate the axes to transform the equation 4x^2-sqrt3xy+y^2=5 into a new equation with no xy term and then find the angle of rotation?

Dec 31, 2016

#### Answer:

Here is a good reference on the topic Rotation of Conic Sections. Please see below.

#### Explanation:

The general Cartesian form for the equation of a conic section is:

$A {x}^{2} + B x y + C {y}^{2} + D x + E y + F = 0$

Equation [9.4.6] in the reference gives us the angle of rotation:

$\theta = \frac{1}{2} {\tan}^{-} 1 \left(\frac{B}{C - A}\right) \text{ [9.4.6]}$

In given equation, $A = 4 , B = - \sqrt{3} \mathmr{and} C = 1$

$\theta = \frac{1}{2} {\tan}^{-} 1 \left(\frac{\sqrt{3}}{3}\right)$

$\theta = \frac{\pi}{12}$ radians

Note: This is the angle of rotation that makes the $x y$ term become zero and, thereby, makes the graph become "un-rotated". The graph of the equation that contains the $x y$ will appear to be rotated by the same amount in the opposite direction.

To find the un-rotated value for A, use equation [9.4.4a]:

A' = (A + C)/2 + [(A - C)/2] cos 2θ - B/2 sin 2θ

$A ' \approx 4.33$

We know that $B '$ will be zero but you may use equation [9.4.4b], if you like.

To find the un-rotated value for C, use the equation [9.4.4c]:

C' = (A + C)/2 + [(C - A)/2] cos 2θ + B/2 sin 2θ

$C ' \approx 0.77$

Equation [9.4.4f] tells us that the constant is unchanged.

Here is the un-rotated equation:

$4.33 {x}^{2} + 0.77 {y}^{2} = 5$

Here is a graph of both equations: