Question

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=2 \cos \theta, \quad y=6 \sin \theta $$

Short answer

The corresponding rectangular equation of the parametric equations \(x = 2 \cos \theta\) and \(y = 6 \sin \theta\) is \((x/2)^2 + (y/6)^2 = 1\). The curve is an ellipse oriented counterclockwise with semi-axes of 2 and 6.

Step-by-step solution

Understand the parametric equations

Given: \(x = 2 \cos \theta\) and \(y = 6 \sin \theta\). Each value of \(\theta\) corresponds to a point \((x, y)\) on the curve. The orientation of the curve is counterclockwise, as the parameter \(\theta\) increases from 0 to 2\(\pi\).

Sketch the curve

To draw the curve, begin at the point corresponding to \(\theta = 0\) and proceed counterclockwise. \(\theta = 0\) gives \((x, y) = (2, 0)\), \(\theta = \pi/2\) gives \((x, y) = (0, 6)\), \(\theta = \pi\) gives \((x, y) = (-2, 0)\), and \(\theta = 3\pi/2\) gives \((x, y) = (0, -6)\). Joining these points in a smooth curve gives an ellipse oriented counterclockwise with semi-axes of 2 and 6.

Eliminate the parameter

To get the rectangular equation, we can eliminate the parameter \(\theta\) from the equations \(x = 2 \cos \theta\) and \(y = 6 \sin \theta\). Observing these equations, it can be seen that they resemble the equations for an ellipse in parametric form, \(x = a \cos \theta\) and \(y = b \sin \theta\), where \(a\) and \(b\) are the semi-major and semi-minor axes. With \(a = 2\) and \(b = 6\), squaring and adding these equations results in \((x/2)^2 + (y/6)^2 = 1\). Thus the rectangular equation of the ellipse is \((x/2)^2 + (y/6)^2 = 1\).