Tom Cooper

If we rotate a point (a,b) about the origin by degrees, then equations relating it to its image are

y = -a sin() + b cos()

LetÕs apply this with = 45^{o} to x^{2} + nxy + y^{2} = 9.

x^{2} + nxy + y^{2} = 9

Now we can see that this is an ellipse centered at the origin for Š2<n<2.

It is a circle of radius 3, centered at (0,0), when n = 2, and it is a hyperbola centered at the origin when n<-2 or n>2.