Problem 18 - Contemporary Precalculus

Question

In Exercises \(15-24,\) use the techniques of Examples 4 and 5 to graph the equation in a suitable square viewing window. $$25(x-5)^{2}+36(y+4)^{2}=900$$

Step-by-Step Solution

Answer

Answer: The ellipse has a center at (5, -4), a major-axis of 12, and a minor-axis of 10.

Step 1: Identify the equation type

The given equation is in the form of an ellipse because it has both x and y squared terms and their coefficients have different signs followed by an equal sign and a constant.

Step 2: Write the equation in standard form for ellipse

To write the given equation in the standard form for ellipse, divide both sides by the constant on the right side of the equation. The standard form for ellipse: \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\). After dividing, the equation becomes: $$\frac{(x-5)^{2}}{36}+\frac{(y+4)^{2}}{25}=1$$

Step 3: Identify the center, major axis, and minor axis

From the standard form of the ellipse equation, we can identify the center as \((h, k) = (5, -4)\). Since \(a^2 = 36\) and \(b^2 = 25\), we have \(a = 6\) and \(b=5\). The major axis is \(2a=12\) and the minor axis is \(2b=10\).

Step 4: Graph the ellipse

To graph the ellipse, follow these steps: 1. Mark the center \((5, -4)\) on the graph 2. From the center, move \(a = 6\) units right and left, which will give the points \((5\pm6, -4)\), these are the vertices of the ellipse. 3. From the center, move \(b = 5\) units up and down, which will give the points \((5, -4\pm5)\), these are the co-vertices of the ellipse. 4. Connect the vertices and co-vertices with a smooth curve, which will represent the ellipse. Now, the equation \(25(x-5)^{2}+36(y+4)^{2}=900\) has been graphed as an ellipse with center \((5,-4)\), major-axis \(12\) and minor-axis \(10\).

Key Concepts

Graphing ellipses

In mathematics, graphing an ellipse involves plotting a smooth, oval-shaped curve. An ellipse is defined by its equation, and it's essential to understand the components involved to draw its graph accurately. When graphing an ellipse like the one given in the exercise, start by locating the **center** of the ellipse on a coordinate plane. This center point acts as the midpoint and is key in positioning the ellipse.

• First, use the coordinates of the center, which are given by points \(h\) and \(k\) from the standard equation form.
• Next, determine how far you need to move horizontally and vertically to place the ellipse's vertices and co-vertices.
• In our exercise, the major and minor axes help in stretching the ellipse to its correct size.
• Plot these key points and draw a smooth curve connecting them to finish the graph of the ellipse.

Standard form of ellipse

To graph an ellipse correctly, converting its equation to the standard form of an ellipse is crucial. This standard form looks like \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]and allows easy identification of essential features like the center and axes.

• **Center (h,k):** From the transformed equation, the center can be extracted directly as \(h,k\).
• **Value of \(a\) and \(b\):** Dividing by the constant transforms the equation into the required form, revealing \(a\) and \(b\) as the square roots of the divisors of squared terms.
• Knowing whether \(a > b\) helps in determining the major axis direction, which is crucial to identifying the ellipse's orientation.

Elliptical equations

Elliptical equations have unique properties that differentiate them from other conical sections like circles or hyperbolas. The distinguishing factor is the presence of different coefficients in the squared terms of \(x\) and \(y\), yet both remain positive.

• **Expression of Differences:** The equation maintains a constant form where variations in lengths along horizontal and vertical directions are evident.
• **Formation:** These equations usually require simplification (by dividing all terms by a constant) to garner useful information about the ellipse.
• Unlike circles, which have equal radii, ellipses will have different lengths for its axes, defined by the equation's coefficients.

Axes of ellipse

Understanding the axes of an ellipse is key to properly graphing and analyzing its shape. An ellipse has two main axes: the major and the minor axis.

• **Major Axis:** This is the longest axis and stretches across the ellipse. Its length is determined by \(2a\), which comes from twice the value of \(a\) in the standard form equation.
• **Minor Axis:** This axis is shorter, stretching across the width of the ellipse, determined by \(2b\), which is twice the value of \(b\).
• The axes intersect at the ellipse’s center, and they are perpendicular to each other, providing symmetry to the shape.
• For the ellipse we're working with, the major axis would span 12 units, while the minor axis spans 10 units, showing how these axes define the overall dimension of the ellipse.