Question

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

Answer 1

The conic section is a hyperbola

Answer 2

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

Now, we know that
`csc(t)=(1)/(sin(t))` and
`cot(t)=(cos(t))/(sin(t))`. A nice look at the given parametric equations tells that:

`x^2-y^2=csc^2(t)-cot^2(t)=(1)/(sin^2(t))-(cos^2(t))/(sin^2(t))=(1-cos^2(t))/(sin^2(t))=(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.