How do I parameterize $Ax^2+Bxy+Cy^2=1$?

Question

So I have a point that moves along a curve at a constant speed: $$Ax^2+Bxy+Cy^2=1$$ I have to find velocity and acceleration. To do this I think I need to find $x(t)$ and $y(t)$. That is, unfortunately, as far as I've gotten with this problem. I'm not even sure what shape is being formed exactly, is it an ellipse? Can anyone point me in the right direction? I would be very grateful for any help

Answer 1

Hint

Not sure that you have to parameterize the curve.

If a point $(x_0,y_0)$ lies on the curve, you have $Ax_0^2 +Bx_0y_0 +Cy_0^2-1=0$. You are looking for $(x_0^\prime,y_0^\prime)$ where the derivative is taken with regard to the arc length parameterization. Therefore you also have $(x_0^\prime)^2 +(y_0^\prime)^2=1$.

Taking the derivative of the first equation, you get another equation involving $x_0^\prime, y_0^\prime$.

Solving those two equations should lead you to $(x_0^\prime,y_0^\prime)$.

Not fun computation... but should lead to the result. And then another round to get the acceleration.