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?

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Answer 1 · 2017-10-25T00:25:44

See below.

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

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

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