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?

1 Answer
Oct 25, 2017

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):}#