Question

Find parametric equations (not unique) of the following ellipses (see Exercises \(71-72\) ). Graph the ellipse and find a description in terms of \(x\) and \(y\) An ellipse centered at (0,-4) with major and minor axes of lengths 10 and \(3,\) on the \(x\) - and \(y\) -axes, respectively, generated clockwise (Hint: Shift the parametric equations.)

Short answer

Question: Find the parametric equations for an ellipse centered at (0, -4) with major and minor axes of lengths 10 and 3, respectively, on the x- and y-axes. Then, graph the ellipse and find a description in terms of x and y. Answer: The parametric equations for the ellipse are \(x(t) = 10\cos(-t)\) and \(y(t) = -4 + 3\sin(-t)\). The description in terms of x and y is \(\frac{x^2}{100} + \frac{(y+4)^2}{9} = 1\).

Step-by-step solution

Write the general equation for an ellipse

To find the parametric equations, we first need to recall the general equation for an ellipse with a center at (h, k), major axis a, and minor axis b: \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\) In our case, the center is (0, -4), a = 10, and b = 3, so the equation for the ellipse becomes: \(\frac{x^2}{100} + \frac{(y+4)^2}{9} = 1\)

Find the parametric equations for the ellipse

Parametric equations can be written as a pair of functions for x(t) and y(t), where t is a parameter. For an ellipse, we use: \(x(t) = h + a\cos(t)\) \(y(t) = k + b\sin(t)\) Since the ellipse is generated clockwise, we'll use negative t for the parameter. So, for our ellipse, the parametric equations become: \(x(t) = 10\cos(-t)\) \(y(t) = -4 + 3\sin(-t)\)

Graph the ellipse and find a description in terms of x and y

To graph the ellipse, you can plug different values of t in x(t) and y(t) and plot the corresponding points on a coordinate plane (you can use a graphing calculator or online tool for this). The description in terms of x and y is the equation we found in Step 1: \(\frac{x^2}{100} + \frac{(y+4)^2}{9} = 1\) So the final solution is: Parametric equations: \(x(t) = 10\cos(-t)\) \(y(t) = -4 + 3\sin(-t)\) Description in terms of x and y: \(\frac{x^2}{100} + \frac{(y+4)^2}{9} = 1\)

Key concepts

Ellipse

An ellipse is a special type of oval shape that can be described by a specific mathematical equation. In an ellipse, there are two main axes: the major axis and the minor axis. The major axis is the longest diameter, while the minor axis is the shortest.
To describe an ellipse mathematically, we use the standard form equation: \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\]

• Where \(h, k\) is the center of the ellipse.
• \(a\) is the length of the semi-major axis.
• \(b\) is the length of the semi-minor axis.

The axes can either be horizontal or vertical, and their orientation dictates how you should place \(a\) and \(b\) in the equation. In the given example, the ellipse is centered at \( (0, -4) \), with the major axis on the x-axis length 10 (so \(a=10\)) and the minor axis on the y-axis length 3 (so \(b=3\)). You place these in the equation to describe the ellipse: \[\frac{x^2}{100} + \frac{(y+4)^2}{9} = 1\]

Trigonometric Functions

Trigonometric functions such as sine and cosine are key tools when dealing with parametric equations of an ellipse. These functions help us describe the position of points on the ellipse over a parameter, commonly represented by \(t\).
In parametric equations:

• \(x(t) = h + a\cos(t)\)
• \(y(t) = k + b\sin(t)\)

Using cosine and sine ensures that the movement along the ellipse is smooth and continuous. Each value of \(t\) points to a specific location on the ellipse. These equations can change direction based on how \(t\) is defined. For instance, if you want the ellipse to be traced clockwise, you could use negative \(t\). In the provided problem, the parametric equations were given as:

• \(x(t) = 10\cos(-t)\)
• \(y(t) = -4 + 3\sin(-t)\)

These equations will trace the ellipse clockwise around its center at \( (0, -4) \).

Conic Sections

Conic sections are shapes that can be formed by cutting a double-napped cone with a plane. These sections include ellipses, parabolas, circles, and hyperbolas, with each shape having unique properties.
An ellipse emerges when a plane intersects the cone at an angle that is oblique to its axis, and not parallel to the base. The general form for conics is given by a quadratic equation in two variables:\[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\]Ellipses, as part of conic sections, share relationships with other shapes in this family. They represent bounded curves, meaning all points are contained within a specific region.
Parametric equations are one way to explore their geometric properties and give practical methods to graph these curves. In this problem specifically, understanding the position and orientation of the ellipse helped determine these parameters: \(a and b\) in the parametric and standard forms. By setting and shifting these parameters appropriately, the ellipse can be drawn accurately from the origin point to its center at \( (0, -4) \).