Question

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?

Answer 1

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:

`Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0`

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

`theta = 1/2tan^-1(B/(C - A))" [9.4.6]"`

In given equation, `A = 4, B = -sqrt(3) and C = 1`

`theta = 1/2tan^-1(sqrt(3)/3)`

`theta = pi/12` radians

Note: This is the angle of rotation that makes the `xy` term become zero and, thereby, makes the graph become "un-rotated". The graph of the equation that contains the `xy` 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' ~~ 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' ~~ 0.77`

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

Here is the un-rotated equation:

`4.33x^2 + 0.77y^2 = 5`

Here is a graph of both equations: