Answer
$x^2=5(y-5)$
Work Step by Step
We are given the parametric equations:
$\begin{cases}
x=5\tan t\\
y=5\sec^2 t
\end{cases}$
with $-\dfrac{\pi}{2}\lt t\lt \dfrac{\pi}{2}$
Rewrite the equations:
$\begin{cases}
\tan t=\dfrac{x}{5}\\
\sec^2 t=\dfrac{y}{5}
\end{cases}$
Use the identity:
$\tan^2 t+1=\sec^2 t$
$\left(\dfrac{x}{5}\right)^2+1=\dfrac{y}{5}$
$\dfrac{x^2}{25}+1=\dfrac{y}{5}$
$x^2+25=5y$
$x^2=5y-25$
$x^2=5(y-5)$
The curve represents a parabola.
