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Section 3-1 : Parametric Equations and Curves

14. Write down a set of parametric equations for the following equation.

\[\frac{{{x^2}}}{4} + \frac{{{y^2}}}{{49}} = 1\]

The parametric curve resulting from the parametric equations should be at \(\left( {0, - 7} \right)\) when \(t = 0\) and the curve should have a clockwise rotation.

Show Solution

If we don’t worry about the “starting” point (i.e. where the curve is at when \(t = 0\)) and we don’t worry about the direction of motion we know from the notes that the following set of parametric equations will trace out the ellipse given by the equation above.

\[\begin{align*}x & = 2\cos \left( t \right)\\ y & = 7\sin \left( t \right)\end{align*}\]

The problem with this set of parametric equations is that when \(t = 0\) we are at the point \(\left( {7,0} \right)\) which is not the point we are supposed to be at. Also, from our knowledge of the examples worked in the notes for this section or an analysis similar to some of the earlier problems in this section we can see that the parametric curve traced out by this set of equations will trace out in a counter clockwise rotation – again not what we need.

So, we need to come up with a different set of parametric equations that meets the requirements.

The first thing to acknowledge is that using sine and cosine will always be the easiest way to get a set of parametric equations for an ellipse. However, there is no reason at all to always use cosine for the \(x\) equation and sine for the \(y\) equation.

Knowing that we need \(x = 0\) and \(y = - 7\) when \(t = 0\) and using the fact that we know that \(\sin \left( 0 \right) = 0\) and \(\cos \left( 0 \right) = 1\) the following set of parametric equations will “start” at the correct point when \(t = 0\).

\[\begin{align*}x & = - 2\sin \left( t \right)\\ y & = - 7\cos \left( t \right)\end{align*}\]

All we need to do now is check if this will trace out the ellipse in a clockwise direction.

If we start at \(t = 0\) and increase \(t\) until we reach \(t = \frac{\pi }{2}\) we know that sine will increase from 0 to 1. This will in turn mean that \(x\) must decrease (don’t forget the minus sign on the \(x\) equation) from 0 to -2.

Likewise, increasing \(t\) from \(t = 0\) to \(t = \frac{\pi }{2}\) we know that cosine will decrease from 1 to 0. This in turn means that \(y\) will increase (don’t forget the minus sign on the \(y\) equation!) from -7 to 0.

The only way for both of these things to happen at the same time is for the curve to start at \(\left( {0, - 7} \right)\) when \(t = 0\) and trace along the ellipse in a clockwise direction until we reach the point \(\left( { - 2,0} \right)\) when \(t = \frac{\pi }{2}\).

We could continue in this fashion further increasing \(t\) until it reaches \(t = 2\pi \) (which will put us back at the “starting” point) and convince ourselves that the ellipse will continue to trace out in a clockwise direction.

Therefore, one possible set of parametric equations that we could use is,

\[\require{bbox} \bbox[2pt,border:1px solid black]{\begin{align*}x & = - 2\sin \left( t \right)\\ y & = - 7\cos \left( t \right)\end{align*}}\]

We’ll leave this problem with a final note about the answer here. This is possibly the “simplest” answer we could give but it is completely possible that you may have come up with a different answer to this problem. There are almost always lots of different possible sets of parametric equations that will trace out a particular parametric curve according to some particular set of restrictions.