Problem 52 - 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, and ellipses. \(16(x-5)^{2}+25(y-4)^{2}=400\)

Step-by-Step Solution

Answer

The equation in standard form is \(\frac{(x-5)^2}{25} + \frac{(y-4)^2}{16} = 1\), and it represents an ellipse centered at (5, 4), with semi-major axis 5 and semi-minor axis 4.

Step 1: Understanding the Equation

The given equation is of the form \( 16(x-5)^2 + 25(y-4)^2 = 400 \). This resembles the general equation of an ellipse: \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \). The goal is to rewrite this equation in standard form.

Step 2: Converting to Standard Form

Divide each term in the equation \( 16(x-5)^2 + 25(y-4)^2 = 400 \) by 400: \[ \frac{16(x-5)^2}{400} + \frac{25(y-4)^2}{400} = 1 \]. Simplifying gives \[ \frac{(x-5)^2}{25} + \frac{(y-4)^2}{16} = 1 \]. This is the standard form of an ellipse.

Step 3: Identifying Elements of the Ellipse

In the standard form \( \frac{(x-5)^2}{25} + \frac{(y-4)^2}{16} = 1 \), \(a^2 = 25\) and \(b^2 = 16\). Therefore, \(a = 5\) and \(b = 4\). The center of the ellipse is \((h, k) = (5, 4)\). The major axis is along the x-axis since \(a > b\).

Step 4: Graphing the Ellipse

To graph the ellipse, plot the center at \((5, 4)\). From the center, move 5 units left and right along the x-axis (major axis), and move 4 units up and down along the y-axis (minor axis) to find the vertices and co-vertices. Draw a smooth, oval shape through these points to complete the ellipse.

Key Concepts

Converting to Standard Form

Transforming an equation of an ellipse into its standard form is a crucial step in understanding its geometric properties. Let's delve into how this conversion is achieved using a simple algebraic manipulation.
The standard form of an ellipse is \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]This form allows us to easily identify the center, vertices, and axes of the ellipse. The given equation, \( 16(x-5)^2 + 25(y-4)^2 = 400 \), is not in this standard form, as the right side is not equal to 1.
By dividing each term by 400, our goal is to adjust the equation:

• Divide by 400: \( \frac{16(x-5)^2}{400} + \frac{25(y-4)^2}{400} = 1 \)
• Simplify: \( \frac{(x-5)^2}{25} + \frac{(y-4)^2}{16} = 1 \)

Now, the equation is in the standard form with center at \((h, k) = (5, 4)\).
This transformation makes it straightforward to identify and work with the ellipse's properties.

Graphing Ellipses

Graphing an ellipse involves understanding its core elements and positioning it correctly on the coordinate plane. Let's explore how to graph the ellipse, now that it is in standard form. Given the equation:\[ \frac{(x-5)^2}{25} + \frac{(y-4)^2}{16} = 1 \]First, identify the center of the ellipse, which is at \((5, 4)\), derived from the values of \(h\) and \(k\). Placing a point here serves as your starting position.

Next, determine the lengths of the semi-major and semi-minor axes:

• \(a^2 = 25 \rightarrow a = 5\)
• \(b^2 = 16 \rightarrow b = 4\)

The larger value, \(a = 5\), indicates that the ellipse is longer along the x-axis. To plot the ellipse:

• From the center \((5, 4)\), move 5 units left and right along the x-axis for the vertices.
• Move 4 units up and down along the y-axis for the co-vertices.
• Draw an oval shape passing through these critical points.

This method converts the equation into a visual representation, making the abstract more concrete.

Geometry of Ellipses

The geometry of ellipses is a fascinating subject, as it inherently involves various mathematical properties and relationships. Understanding this geometry helps in predicting and recognizing the shape's behavior. An ellipse can be thought of as the set of all points where the sum of the distances from two different points, called the foci, is constant.

When working with the standard form \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), we have two axes:

• The major axis stretches along the longer dimension (in our case, the x-axis), which is 2\(a\) units long.
• The minor axis, positioned along the shorter dimension (here, the y-axis), is 2\(b\) units long.

The lengths of these axes directly relate to how stretched the ellipse appears.

If \(a > b\), the ellipse is oriented horizontally, while \(b > a\) leads to a vertical orientation. This specific ellipse is horizontal, as \(a = 5\) exceeds \(b = 4\). Knowing the position of the foci, which are \(c\) units away from the center along the major axis, aids in further analysis. Here, you'll often use the relationship \(c^2 = a^2 - b^2\) to find the exact location of the foci.Knowing these geometrical insights sharpens your understanding and skill in handling ellipse-related problems.