Question

How do you rotate the axes to transform the equation `xy+4=0` into a new equation with no xy term and then find the angle of rotation?

Answer 1

See below.

Explanation:

Defining `R_theta = ((Cos theta, -Sintheta),(Sintheta, Costheta))`

and choosing a new coordinate system

`((X),(Y)) = R_theta((x),(y))`

and substituting into

`f(x,y) = x y +4=0`

we arrive at

#X Y (Cos^2theta-sin^2 theta) - X^2 Cos theta Sintheta + Y^2 Costheta Sin[theta -4 = 0#

Now choosing `theta` such that

`Cos^2theta-sin^2 theta = 0`

we get at `theta = pm (3pi)/4` or `theta = pm pi/4`

after this rotation `x y +4 = 0` becomes one of those

`{(4 - X^2/2 + Y^2/2=0),(4 + X^2/2 - Y^2/2=0):}`