Question

Multiple Choice Which of the following describes the graph of the parametric curve \(x=3 t, y=2 t, t \geq 1 ? \mathrm{}\) (A) circle (B) parabola (C) ellipse (D) hyperbola (E) line

Short answer

(E) line

Step-by-step solution

Understand the Parameter's Function

The parameter \(t\) in the curve equations \(x = 3t\) and \(y = 2t\) is the same, meaning the curve is a function of a single parameter. This results in a single, unique location on the curve for each value of \(t\). When \(t\) takes on an increasing set of real numbers (because \(t \geq 1\)), \(x\) and \(y\) also increase.

Explore a Relation between x and y

Express \(y\) in terms of \(x\) to eliminate the parameter \(t\). Using the given parametric equations, we set \(t = x/3\) and \(t = y/2\). Hence, \(x/3 = y/2\). Solving this equation for \(y\), we get \(y = \frac{2}{3}x\), which is a linear equation.

Draw the Graph

Plot the equation \(y = \frac{2}{3}x\) using a graphing tool or by hand, ensuring that the graph is restricted to \(t \geq 1\) or equivalently \(x \geq 3\). As observed, the graph is a straight line, which raises in the positive direction. Hence, the graph is a line segment. In retrospect, the restriction \(t \geq 1\) turns the otherwise infinite line into a line segment.