91Ó°ÊÓ

Parametric equations are a pair of equations that describe the \(x\) and \(y\) coordinates of a point in terms of a single parameter, usually denoted by \(t\). This parameter simultaneously defines both the x and y coordinates as functions of \(t\), and can be written in the form: \(x = f(t)\) \(y = g(t)\) where \(f(t)\) and \(g(t)\) are functions that define the relationship between the parameter \(t\) and the \(x\) and \(y\) coordinates.

Consider the following pair of parametric equations: \(x = t^2\) \(y = 2t\) These equations describe the \(x\) and \(y\) coordinates of a point in terms of the parameter \(t\). For each value of \(t\), we can find a unique point \((x, y)\) on the curve.

To see how this pair of parametric equations generates a curve in the \(x y\)-plane, we need to eliminate the parameter \(t\) from the equations so that we have a single equation that relates \(x\) and \(y\). To do this, first solve one of the parametric equations for \(t\) and then substitute this expression into the other parametric equation. For our example, we can solve the second equation for \(t\): \(t = \frac{y}{2}\) Now, substitute this expression for \(t\) into the first equation: \(x = \left( \frac{y}{2} \right)^2\)

Simplify the above equation to get the final equation relating \(x\) and \(y\): \(x = \frac{y^2}{4}\)

The equation \(x = \frac{y^2}{4}\) describes a parabola with its vertex at the origin and facing to the right. This is the curve generated by the given pair of parametric equations in the \(x y\)-plane. As the parameter \(t\) varies, we trace out different points on this curve, which represents the path of the point \((x, y)\) as defined by the functions \(f(t) = t^2\) and \(g(t) = 2t\). In conclusion, a pair of parametric equations generates a curve in the \(x y\)-plane by defining the \(x\) and \(y\) coordinates of a point as functions of a single parameter, \(t\). By eliminating the parameter, we can find the equation of the curve in the \(x y\)-plane, which can be graphed to visualize the path traced out by the point as \(t\) varies.