Question

What conic section is drawn by the parametric equations x = csct and y = cott?

Answer 1

The given parametric equations are$x = csc(t)$ and $y = cot(t)$.

Now, we know that $csc(t)=\frac{1}{sin(t)}$ and $cot(t)=\frac{cos(t)}{sin(t)}$. A nice look at the given parametric equations tells that:

$x^2-y^2=csc^2(t)-cot^2(t)=\frac{1}{sin^2(t)}-\frac{cos^2(t)}{sin^2(t)}=\frac{1-cos^2(t)}{sin^2(t)}=\frac{sin^2(t)}{sin^2(t)}=1$

This is the equation of a Hyperbola.

Thus, the conic section which is drawn by the parametric equations x = csct and y = cott is a hyperbola.

Answer 2

The conic section is a hyperbola