Problem 24 - Precalculus Mathematics for Calculus

Question

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$9 x^{2}-36 x+4 y^{2}=0$$

Step-by-Step Solution

Answer

The equation represents an ellipse centered at (2, 0) with a vertical major axis.

Step 1: Group and Rearrange Terms

Start by grouping the terms involving the same variables together. The equation is given as: \[9x^2 - 36x + 4y^2 = 0\]Rearrange terms grouped by their variables:\[9(x^2 - 4x) + 4y^2 = 0\]

Step 2: Complete the Square for x-terms

To complete the square for the terms \(x^2 - 4x\), follow these steps:1. Take the coefficient of \(x\), which is \(-4\), divide it by 2, and square the result: \[\left(\frac{-4}{2}\right)^2 = 4\]2. Add and subtract this square inside the parentheses: \[9(x^2 - 4x + 4 - 4) + 4y^2 = 0\]Simplify the equation:\[9((x-2)^2 - 4) + 4y^2 = 0\]

Step 3: Simplify the Equation

Distribute and simplify the equation:\[9(x-2)^2 - 36 + 4y^2 = 0\]Add 36 to both sides to isolate the squared terms:\[9(x-2)^2 + 4y^2 = 36\]

Step 4: Identify the Standard Form of an Ellipse

Divide the entire equation by 36 to match the standard form:\[\frac{9(x-2)^2}{36} + \frac{4y^2}{36} = 1\]Simplify to achieve the standard form of an ellipse:\[\frac{(x-2)^2}{4} + \frac{y^2}{9} = 1\]This reveals it's an ellipse centered at \((2, 0)\) with a horizontal minor axis and a vertical major axis.

Step 5: Determine Features of the Ellipse

Identify the center, axes lengths, and other features:1. **Center**: \((2, 0)\)2. **Vertices on the major axis** \((y)\): From the center along \(y\)-axis: \(\pm a = \pm 3\); so vertices are \((2, 3)\) and \((2, -3)\)3. **Vertices on the minor axis** \((x)\): From the center along \(x\)-axis: \(\pm b = \pm 2\); so vertices are \((4, 0)\) and \((0, 0)\)4. **Foci**: Calculate \(c\) using \(c^2 = a^2 - b^2\), where \(a = 3\), \(b = 2\):\[c^2 = 9 - 4 = 5\Rightarrow c = \sqrt{5}\]Foci are \((2, \sqrt{5})\) and \((2, -\sqrt{5})\).

Key Concepts

Ellipse

The ellipse is a fascinating shape that falls under the category of conic sections. Conic sections are curves obtained by intersecting a plane with a cone. An ellipse specifically appears when the intersecting plane cuts through the cone at an angle. Unlike the circle, an ellipse is defined by having two different radii. These are formally known as the major and minor axes. The major axis is the longest diameter of the ellipse, while the minor axis is the shortest. Both axes intersect at the center of the ellipse, which acts as a symmetrical midpoint.

An essential property of the ellipse is its two foci. While a circle has a single center point, an ellipse has two foci. The total distance from any point on the ellipse to these two foci is always constant. This property is unique to ellipses and underscores their symmetrical nature. Real-world examples of ellipses include the orbits of planets around the Sun and the shape of many planetary bodies themselves.

Completing the Square

Completing the square is a valuable technique used in algebra to simplify quadratic equations or to change the form of an equation to make it easier to solve or graph. This method transforms a standard quadratic expression into one that describes a variation of squaring binomials. Doing so makes it easier to analyze properties of the function or conic section, like finding the vertex of a parabola or the center of an ellipse.

Here's a simplified breakdown of how to complete the square:

• Identify the quadratic and linear terms within the equation.
• Take the coefficient of the linear term, divide it by two, and then square it.
• Add and subtract this squared number within the equation to preserve balance.
• Re-factor the newly formed perfect square trinomial, simplifying the expression into a squared binomial form.

This approach is particularly handy for turning standard conic equations into their identifiable forms, allowing us to easily locate key features. In our case, it was used to transform the given equation into an elliptical form.

Standard Form of Ellipse

The standard form of an ellipse is a specialized equation that simplifies the identification of its characteristics, such as the center, axes lengths, and foci. An ellipse centered at the origin has an equation of the form:

• Horizontal orientation: \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \)
• Vertical orientation: \( \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \)

Here, \((h, k)\) is the center of the ellipse, \(a\) represents the semi-major axis's length, and \(b\) represents the semi-minor axis's length. The values of \(a\) and \(b\) follow the relationship \(a > b\) for horizontal ellipses and \(b > a\) for vertical ellipses. The sum of the values under each squared term always equals one, which provides the necessary scaling for their geometric properties.

Converting an equation into this form facilitates the process of graphing the ellipse as well as calculating other valuable properties such as its foci, which are derived using the formula \(c^2 = a^2 - b^2\). Hence, understanding and identifying this form is crucial when working with ellipses in mathematics.