Parametric curves | Mathematico

If we allow $t$ to be a free variable, then we can think about the set of all points $(x, y)$ where

\begin{aligned} x & = f (t) \\ y & = g (t) \end{aligned}

are functions of $t .$

For example, if

\begin{aligned} x & = t^{2} + 2 t + 1 \\ y & = t^{2} - 1 \end{aligned}

then we could make a table of values

\begin{array}{cccccc} t & - 2 & - 1 & 0 & 1 & 2 \\ x & - 1 & - 2 & - 1 & 2 & 7 \\ y & 3 & 0 & - 1 & 0 & 3 \end{array}

So the points

(- 1, 3), (- 2, 0), (- 1, - 1), (2, 0), (7, 3)

lie on the curve.

We could plot these points to find out what the curve would look like:

You should move the slider for $t$ to see how the point $P$ moves, noticing that the $x$ coordinate of $P$ is always $t^{2} + 2 t - 1$ and the $y$ coordinate is always $t^{2} - 1.$

The curve $C$ is given parametrically by the functions

\begin{aligned} x & = t^{2} - 2 t - 3 \\ y & = t^{2} + 2 t \end{aligned}

Find the equation of the tangent to $C$ at the point $A$ .
Find the coordinates of the point $B$ .
Show that points on the curve $C$ satisfy the implicit Cartesian equation
$x^{2} = (y + 1) (2 x - y + 3)$