Question

Parametric equations of ellipses Find parametric equations (not unique) of the following ellipses. Graph the ellipse and find a description in terms of \(x\) and \(y\). An ellipse centered at the origin with major axis of length 6 on the \(x\) -axis and minor axis of length 3 on the \(y\) -axis, generated counterclockwise

Short answer

Question: Given an ellipse with a major axis length of 6 and a minor axis length of 3, write its parametric equations and describe the ellipse in terms of x and y. Answer: The parametric equations of the ellipse are x = 6 * cos(t) and y = 3 * sin(t). The ellipse can be described in terms of x and y using the equation (x^2 / 36) + (y^2 / 9) = 1.

Step-by-step solution

Write parametric equations of the ellipse

Given the major axis length 6 and minor axis length 3, the parametric equations of the ellipse are: $$x = 6 \cdot \cos(t)$$ $$y = 3 \cdot \sin(t)$$

Graph the ellipse and describe it in terms of x and y

To eliminate the parameter 't' from the parametric equations, we can use the trigonometric identity \(\sin^2(t) + \cos^2(t) = 1\). Divide the \(x\) equation by 6: $$\frac{x}{6} = \cos(t)$$ Square both sides: $$\frac{x^2}{36} = \cos^2(t)$$ Divide the \(y\) equation by 3: $$\frac{y}{3} = \sin(t)$$ Square both sides: $$\frac{y^2}{9} = \sin^2(t)$$ Add both equations: $$\frac{x^2}{36} + \frac{y^2}{9} = \cos^2(t) + \sin^2(t)$$ Now, we use the trigonometric identity \(\sin^2(t) + \cos^2(t) = 1\) to finalize the description of the ellipse in terms of \(x\) and \(y\): $$\frac{x^2}{36} + \frac{y^2}{9} = 1$$ The graph of the ellipse will be centered at the origin with a major axis of length 6 along the \(x\)-axis and a minor axis of length 3 along the \(y\)-axis. The ellipse is generated counterclockwise.

Key concepts

Ellipses

Ellipses are smooth, closed curves that resemble elongated circles. They are defined by two main axes - the major and minor axes. These are the longest and shortest diameters of the ellipse, respectively. An ellipse can be centered at any point on a graph, but here, we focus on ellipses centered at the origin \(0, 0\). The standard equation for an ellipse centered at the origin is \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\], where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes. The major axis is along the \(x\)-axis if \(a > b\), and along the \(y\)-axis if \(b > a\). For example, in our given problem, an ellipse with a major axis of length 6 and a minor axis of length 3 can be described by \[\frac{x^2}{6^2} + \frac{y^2}{3^2} = 1\]. This particular ellipse is aligned with the \(x\)-axis as its major axis.

Trigonometric Identities

Trigonometric identities are equations that are always true for any value of the variable. They are especially useful for simplifying expressions and solving equations. One of the most fundamental trigonometric identities is \(\sin^2(t) + \cos^2(t) = 1\). In the context of parametric equations of an ellipse, this identity helps derive the standard form of the ellipse by eliminating the parameter \(t\). By breaking down the ellipse's parametric equations to \(x = a \cdot \cos(t)\) and \(y = b \cdot \sin(t)\), we divide and square each equation, then utilize this identity to transition from parametric to rectangular coordinates:

• Divide \(x = 6 \cos(t)\) by 6, yielding \(\frac{x}{6} = \cos(t)\), and square to obtain \(\frac{x^2}{36} = \cos^2(t)\).
• Similarly, divide \(y = 3 \sin(t)\) by 3, giving \(\frac{y}{3} = \sin(t)\), and square to find \(\frac{y^2}{9} = \sin^2(t)\).

Add these to satisfy \(\sin^2(t) + \cos^2(t) = 1\), resulting in \(\frac{x^2}{36} + \frac{y^2}{9} = 1\), describing the ellipse without the parameter \(t\).

Graphing

Graphing an ellipse involves plotting the curve based on its parametric or rectangular equation, ensuring it reflects the defined major and minor axes. The position \(\left( a \cos(t), b \sin(t) \right)\) moves counterclockwise from the point \(a, 0\) as \(t\), the parameter, changes. For the given problem,

• First identify the lengths of axes using equations \(x = 6 \cos(t)\) and \(y = 3 \sin(t)\). Each cycle from \(t = 0 \ to \ 2\pi\) traces a complete ellipse.
• The graph starts from the positive end of the x-axis when \(t = 0\) (since \(x = 6\ cos(0)\) is maximum and \(y = 0\)).

The process repeats in the second quadrant as \(t\) progresses to \(rac{\pi}{2}\), moving to the negative y-direction, eventually retuning at \(t = 2\pi\) at its start, completing the counterclockwise cycle. The graph clearly shows an elongated shape along the x-axis with endpoints at \(-6, 0\) and \(6, 0\), and along the y-axis at \(0, -3\) and \(-0, 3\). Displaying the ellipse, you measure symmetry and correctness against this theoretical model, visually confirming its orientation and dimensions.