Problem 27 - A Graphical Approach to Precalculus with Limits

Question

Find an equation for each ellipse. \(x\) -intercepts \((\pm 4,0) ; y\) -intercepts \((0, \pm 2)\)

Step-by-Step Solution

Answer

The equation of the ellipse is \(\frac{x^2}{16} + \frac{y^2}{4} = 1\).

Step 1: Understand the Problem Context

We need to find the equation of an ellipse. The problem provides the x-intercepts as \((\pm 4, 0)\) and the y-intercepts as \((0, \pm 2)\). This information helps us to identify that the ellipse is centered at the origin with given semi-major and semi-minor axes lengths.

Step 2: Identify the Semi-Major and Semi-Minor Axes

Given the x-intercepts \((\pm 4, 0)\), these points lie on the x-axis. Therefore, the semi-major axis has a length of 4 along the x-axis. The y-intercepts are \((0, \pm 2)\), indicating that the semi-minor axis has a length of 2 along the y-axis.

Step 3: Write the Standard Form of the Ellipse Equation

For an ellipse centered at the origin, the equation 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. From the intercepts, \(a = 4\) and \(b = 2\).

Step 4: Substitute the Values of a and b into the Equation

Using \(a = 4\) and \(b = 2\), the equation becomes \(\frac{x^2}{4^2} + \frac{y^2}{2^2} = 1\). Simplify this to \(\frac{x^2}{16} + \frac{y^2}{4} = 1\).

Step 5: Confirm the Equation Satisfaction with the Intercepts

The derived equation \(\frac{x^2}{16} + \frac{y^2}{4} = 1\) should satisfy the given intercepts. Substituting \(x = 4\) and \(y = 0\) into the equation yields \(\frac{4^2}{16} + 0 = 1\). Similarly, substituting \(x = 0\) and \(y = 2\) gives \(0 + \frac{2^2}{4} = 1\). Both intercept tests are satisfied.

Key Concepts

Understanding x-intercept in Ellipses

The x-intercept of an ellipse is where the ellipse crosses the x-axis. For an ellipse centered at the origin, the x-intercepts are given as \((\pm a, 0)\). In our exercise, the x-intercepts are located at \((+4, 0)\) and \((-4, 0)\). This tells us that the distance from the center of the ellipse to the point where it crosses the x-axis is 4 units on both sides.

• The x-intercepts can help determine the semi-major axis if it is aligned with the x-axis.
• In cases where the x-intercepts are explicitly given, they reveal one key length of the ellipse.

Understanding x-intercepts is important as they provide insight into one of the key characteristics of the ellipse's shape and size. They also help to confirm the accuracy of an ellipse's equation by checking whether they satisfy the equation.

Exploring y-intercept in Ellipses

The y-intercept is where the ellipse intersects the y-axis. These points are found at \((0, \pm b)\) for an ellipse centered at the origin. In this problem, the y-intercepts are given as \((0, +2)\) and \((0, -2)\). This indicates that the ellipse crosses the y-axis at 2 units above and below the center.

• These intercepts can help reveal the semi-minor axis if it is aligned with the y-axis.
• Y-intercepts are just as crucial as x-intercepts in understanding an ellipse's dimensions.

Confirming the y-intercepts within an ellipse equation further assures that the ellipse is accurately represented in its standard form. They are substantial in forming the equation of the ellipse accurately.

Differentiating Semi-Major Axis in Ellipse

The semi-major axis is one of the defining lengths of an ellipse. It is the longest radius that extends from the center of the ellipse to its edge. In mathematical terms, if the x-intercepts of an ellipse are wider than the y-intercepts, the semi-major axis will be found along the x-axis.

• In the given exercise, the x-intercepts \((\pm 4,0)\) determine the semi-major axis, indicated by the value \(a = 4\).
• The length of the semi-major axis is vital in creating the standard form of an ellipse equation.

The semi-major axis is essential for defining the shape of an ellipse, dictating whether the ellipse is wider or taller. It contributes to the standardized ellipse equation as \(\frac{x^2}{a^2}\). Knowing this helps in determining the width and overall dimensions of the ellipse.

Identifying Semi-Minor Axis in Ellipse

The semi-minor axis of an ellipse is the shortest radius from the center to the ellipse's edge. When the y-intercepts of an ellipse are shorter than the x-intercepts, the semi-minor axis lies along the y-axis.

• For this exercise, the y-intercepts \((0, \pm 2)\) indicate the semi-minor axis, with a value of \(b = 2\).
• It is used in the ellipse equation as \(\frac{y^2}{b^2}\).

This semi-minor axis is crucial in the formation of the ellipse's equation to ensure its dimensions are correctly represented. Like the semi-major axis, it helps define whether the ellipse is more circular or elongated in shape. Understanding this concept ensures proper comprehension of the ellipse’s structural properties and their impact on the standard equation form.