Problem 37 - Precalculus: Functions and Graphs

Question

Show that $$x=a \cos t+h, \quad y=b \sin t+k ; \quad 0 \leq t \leq 2 \pi$$ are parametric equations of an ellipse with center \((h, k)\) and axes of lengths \(2 a\) and \(2 b\).

Step-by-Step Solution

Answer

The parametric equations describe an ellipse centered at \((h, k)\) with axes lengths \(2a\) and \(2b\).

Step 1: Understand the Parametric Equations

Given parametric equations are \(x = a \cos t + h\) and \(y = b \sin t + k\), with \(0 \leq t \leq 2\pi\). These equations describe a path where \(x\) and \(y\) vary with \(t\), potentially outlining a shape.

Step 2: Eliminate Parameter t

To show the shape of the path, eliminate the parameter \(t\) by expressing \(\cos t\) and \(\sin t\) in terms of \(x\) and \(y\). Start by rearranging the equations: \(\cos t = \frac{x - h}{a}\) and \(\sin t = \frac{y - k}{b}\).

Step 3: Apply the Pythagorean Identity

Use the identity \(\cos^2 t + \sin^2 t = 1\). Substitute the expressions for \(\cos t\) and \(\sin t\): \( \left(\frac{x - h}{a}\right)^2 + \left(\frac{y - k}{b}\right)^2 = 1 \).

Step 4: Simplify to Ellipse Equation

Recognize that \( \left(\frac{x - h}{a}\right)^2 + \left(\frac{y - k}{b}\right)^2 = 1 \) is the equation of an ellipse centered at \((h, k)\) with semi-major axis \(a\) and semi-minor axis \(b\). The axes lengths are \(2a\) and \(2b\).

Key Concepts

Parametric Equations

In mathematics, parametric equations are used to express a set of related quantities as explicit functions of an independent parameter. For example, the equations \(x = a \cos t + h\) and \(y = b \sin t + k\) describe an ellipse.
These equations are crucial because they break down complex shapes into simple, manageable parts that change with the parameter \(t\). When \(t\) changes from \(0\) to \(2\pi\), the point \((x, y)\) moves around the ellipse's perimeter.

• \(x = a \cos t + h\): Changes provide the horizontal position for a given \(t\)
• \(y = b \sin t + k\): Changes provide the vertical position

In these equations, \(a\) and \(b\) determine the ellipse's size, while \(h\) and \(k\) shift its center to \((h, k)\). This flexibility shows how parametric equations are a powerful tool to describe more than just simple lines or circles.

Axes Lengths

An ellipse has two key parameters known as the axes. The semi-major and semi-minor axes define the shape and extent of the ellipse.

In the parametric equations discussed earlier, \(a\) and \(b\) are directly tied to the lengths of these axes:

• Semi-major axis \(a\): Determines the ellipse's largest radius, running along its longest side.
• Semi-minor axis \(b\): Represents the shortest radius, crossing the semi-major axis.

Therefore, the full lengths of the axes, also known as the diameters, are \(2a\) and \(2b\). This means:

• The ellipse stretches up to \(2a\) units along the x-axis.
• It stretches up to \(2b\) units along the y-axis.

Understanding axes lengths is crucial in engineering and physics, where the dimensions and positioning of objects are often required for practical applications.

Pythagorean Identity

The Pythagorean identity is a well-known trigonometric rule stating that \(\cos^2 t + \sin^2 t = 1\). This identity is pivotal in connecting the parametric equations of an ellipse to its standard algebraic form.

When you utilize this identity with the rearranged versions of the given parametric equations, \(\cos t = \frac{x - h}{a}\) and \(\sin t = \frac{y - k}{b}\), you can derive the equation of an ellipse:

• Substitute: \(\left(\frac{x - h}{a}\right)^2 + \left(\frac{y - k}{b}\right)^2 = 1\)

This equation characterizes an ellipse centered at \((h, k)\), affirming its geometric property. By understanding the Pythagorean identity, it becomes easier to transform parametric forms into standard ones, illustrating how the parameters relate to the shape itself.