Question
How to find the parametric equation of: $x^2-3y^2=1$
I know that $x=\cos{\theta}$ and $x=\sin{\theta}$
But I've come up with the bogus solution that the answer is:
$\cos{\theta}^2-3(\sin{\theta})^2=1$
I was wondering what was the real way in parametricizing this equation?
Best Answer
write $$x^2-3y^2=1$$ $$x^2-(\sqrt{3}y)^2=1$$ compare with $$\cosh^2t-\sinh^2t=1$$ so we have \begin{cases} x=\cosh t\\ \sqrt{3}y=\sinh t \end{cases} or \begin{cases} x=\cosh t\\ y=\dfrac{1}{\sqrt{3}}\sinh t \end{cases}