Question

Find the rectangular equation of the curve.

$x=5\tan t,y=se{c}^{2}t,-\frac{\mathrm{\pi }}{2}<t<\frac{\mathrm{\pi }}{2}$

Short answer

The required equation is$x^2=25(y-1)$

Step-by-step solution

Step 1. Given Information

The given parametric equations are$x=5\tan t,y=5se{c}^{2}t$

Step 2. Explanation

We will use the pythagorean identity, $se{c}^{2}t-{\tan }^{2}t=1$to find the required equation.

Let us obtain the value of $\tan t\mathrm{and}sect$in terms of $x\mathrm{and}y$from the given parametric equations.

localid="1648802898629" role="math" $$\begin{gathered} 5\tan t=x\Rightarrow \tan t=\frac{x}{5} \\ y=se{c}^{2}t\Rightarrow sect=\pm \sqrt{y} \end{gathered}$$

Substitute the values in the pythagorean theorem,

$$\begin{gathered} {\left(\pm \sqrt{y}\right)}^2-{\left(\frac{x}{5}\right)}^2=1 \\ y-{\frac{x}{25}}^2=1 \\ {x}^2=25\left(y-1\right) \end{gathered}$$

Thus, the obtained equation is of parabola whose focal distance is localid="1648803390459" $a=\frac{25}{4}$and whose vertex is at the point with coordinates $\left(0,1\right)$