Problem 42 - Algebra and Trigonometry

Question

Eliminate the parameter. Write the resulting equation in standard form. An ellipse: \(x=h+a \cos t, y=k+b \sin t\)

Step-by-Step Solution

Answer

The standard form of the equation for the given parametric equation of an ellipse is \( \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \).

Step 1: Identify Parametric Equations for an Ellipse

The given parametric equations are \( x = h + a \cos t \) and \( y = k + b \sin t \). Here, \( h \) and \( k \) represent the coordinates of the center of the ellipse, while \( a \) and \( b \) represent the semimajor and semiminor axes, respectively. The parameter \( t \) varies over the interval \( [0, 2\pi] \).

Step 2: Express \( \cos^2 t \) and \( \sin^2 t \) in terms of \( x \) and \( y \)

Isolate \( \cos t \) and \( \sin t \) in terms of \( x \), \( y \), \( h \), \( k \), \( a \), and \( b \). We get:\( \cos t = (x - h) / a \) and \( \sin t = (y - k) / b \). Squaring both sides of each expression gives: \( \cos^2 t = ((x - h) / a)^2 \) and \( \sin^2 t = ((y - k) / b)^2 \).

Step 3: Apply the Pythagorean Identity

Recall that \( \cos^2 t + \sin^2 t = 1 \). Substituting from the previous step: \(((x - h) / a)^2 + ((y - k) / b)^2 = 1 \).

Step 4: Rearrange to Obtain the Standard Form of the Ellipse

Squaring the equations from the previous step gives the standard form of an ellipse centered at \( (h, k) \) with semimajor axis \( a \) and semiminor axis \( b \). Collecting terms, we get: \( \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \). This is the standard form of an ellipse.

Key Concepts

Understanding Parametric Equations

Parametric equations define a set of related quantities as explicit functions of one or more independent variables, known as parameters. In the context of an ellipse, parametric equations help describe the path traced by a point moving around the ellipse with respect to the parameter, usually denoted by \( t \).
For an ellipse centered at \((h, k)\) with parametric equations \( x = h + a \cos t \) and \( y = k + b \sin t \), \( a \) and \( b \) represent the axes of the ellipse. The parameter \( t \) usually ranges from \( 0 \) to \( 2\pi \), covering the entire ellipse.

• \( a \): length of the semimajor axis
• \( b \): length of the semiminor axis
• \( t \): parameter varying over a specific interval

This equation system effectively describes each point on an ellipse as \( t \) changes, offering a dynamic view of the ellipse's shape.

Converting to Standard Form

The standard form of an ellipse provides a more static representation, expressed in terms of \( x \) and \( y \). This form is achieved by eliminating the parameter \( t \) from the parametric equations.
To convert from parametric form to standard form for our ellipse, we solve for \( \cos t \) and \( \sin t \) from the parametric equations:
\( \cos t = \frac{x - h}{a} \) and \( \sin t = \frac{y - k}{b} \)

• Step 1: Square both expressions: \( \cos^2 t = \left(\frac{x - h}{a}\right)^2 \) and \( \sin^2 t = \left(\frac{y - k}{b}\right)^2 \)
• Step 2: Use these in the Pythagorean Identity (next section)

Thus, eliminating \( t \) provides the ellipse's standard form: \[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \]This succinctly captures the ellipse’s geometry.

Exploring the Pythagorean Identity

The Pythagorean Identity is a fundamental trigonometric identity given by \( \cos^2 t + \sin^2 t = 1 \). This identity is crucial for converting parametric equations to the standard form of an ellipse.
When we substitute \( \cos^2 t \) and \( \sin^2 t \) from the parametric equations into this identity, we get:
\[ \left(\frac{x - h}{a}\right)^2 + \left(\frac{y - k}{b}\right)^2 = 1 \]
This formula maintains the geometric properties of the ellipse, adhering to the Pythagorean theorem's principles. It helps link trigonometric components to the classic algebraic expression of an ellipse, showing how the ellipse stretches along its axes while maintaining its closed loop.

• Essential for connecting trigonometric and standard forms
• Provides a reliable way to confirm the integrity of ellipse equations

Understanding this identity helps grasp the transition from parametric to geometric descriptions.