Problem 46 - Intermediate Algebra

Question

Write each equation in standard form, if it is not already so, and graph it. The problems include equations that describe circles, parabolas, ellipses, and hyperbolas. $$ 16 x^{2}+25(y-3)^{2}=400 $$

Step-by-Step Solution

Answer

The equation is an ellipse centered at (0, 3) with a semi-major axis of 5 and semi-minor axis of 4.

Step 1: Identify the Type of Equation

The equation given is \(16x^2 + 25(y-3)^2 = 400\). This equation resembles the general form of the equation of an ellipse, which is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\). The coefficients of \(x^2\) and \((y-3)^2\) indicate it is for an ellipse.

Step 2: Rewrite the Equation in Standard Form

To put the equation in standard form, divide all terms by 400 to set the equation equal to 1. \[ \frac{16x^2}{400} + \frac{25(y-3)^2}{400} = \frac{400}{400} \] Simplifying gives: \[ \frac{x^2}{25} + \frac{(y-3)^2}{16} = 1 \] This is the standard form of the ellipse.

Step 3: Graph the Ellipse

From the standard form \(\frac{x^2}{25} + \frac{(y-3)^2}{16} = 1\), we can determine the center, semi-major axis, and semi-minor axis of the ellipse:- Center: \((h, k) = (0, 3)\)- \(a^2 = 25\) implies \(a = 5\)- \(b^2 = 16\) implies \(b = 4\)Since \(a > b\), the major axis is along the \(x\)-axis.Plot the center at (0,3). Draw the major axis along \(x\) from \(-5\) to \(5\), and the minor axis along \(y\) from \(-1\) to \(7\).

Key Concepts

Ellipse

An ellipse is a type of conic section that looks like an elongated circle, or an oval. It is defined as the set of all points where the sum of the distances from two fixed points, called foci, is constant. This unique property makes them fascinating subjects in both mathematics and real-world applications, such as planetary orbits.

In the context of our equation, the ellipse is described by \(\frac{x^2}{25} + \frac{(y-3)^2}{16} = 1\). The factors of 25 and 16 originate from dealing with the ellipse's major (the longest diameter) and minor (the shortest diameter) axes, which correspond to measures on the x-axis and y-axis, respectively. In general, the greater number between the denominators determines which axis the ellipse is extended along — this is our semi-major axis.

The center of the ellipse is key for locating it on the graph, and is usually written as the point \((h,k)\). For this ellipse, the center is shifted from the origin to \( (0, 3) \). Understanding the position and orientation of the ellipse is crucial when graphing and analyzing its mathematical properties.

Standard Form

The standard form of an equation is a specific way of representing mathematical objects to make comparisons and computations manageable. For ellipses, the standard form helps in quickly recognizing their properties and graphing them accurately. The form \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\) is classic for ellipses, encapsulating their geometric behavior neatly.

In the given solution, rewriting the equation \(16x^2 + 25(y-3)^2 = 400\) by dividing every term by 400 enabled the transformation into standard form. This step is crucial because it normalizes the equation, making it easier to identify the coefficients that determine the length of the axes and the positioning of the center.

• The value \(a\), representing half the length of the major axis, is derived from \(a^2 = 25\), hence \(a = 5\).
• The value \(b\), representing half the length of the minor axis, is derived from \(b^2 = 16\), hence \(b = 4\).

Correctly interpreting an ellipse’s parameters from its standard form allows for clear and precise graphing.

Graphing

Graphing an ellipse involves accurately plotting its principal dimensions - the major and minor axes - and positioning it correctly based on its center. Using the standard form equation, we can graphically display the features of an ellipse.

To start graphing, the center of the ellipse \(h = 0, k = 3\) must be plotted, establishing the starting point. With \((0, 3)\) as the center, the direction of each axis needs to be identified and drawn. Since in our example the \(a\) value, which corresponds to a larger value than \(b\), is associated with the x-component, the major axis lies along the x-axis.

• Mark points along the x-axis 5 units left and right from the center for a total length of 10 units.
• Similarly, mark points along the y-axis 4 units up and down from the center for a total length of 8 units.

This visual representation simplifies understanding of the ellipse's structure and behavior, thus promoting easier interpretation and analysis of conic sections.