Problem 14 - College Algebra

Question

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$9 x^{2}+4 y^{2}=36$$

Step-by-Step Solution

Answer

The graph of the ellipse is a shape centered at the origin (0,0) extending 3 units left and right along the x-axis and 2 units up and down along the y-axis. The foci of the ellipse are located at (-\(\sqrt{5}\),0) and (\(\sqrt{5}\),0).

Step 1: Rewrite the equation in standard form

First, the given equation \(9x^2 + 4y^2 = 36\) should be rewritten in the standard form. This can be done by dividing each term by 36, resulting in the standard form equation \(\frac{x^2}{4} + \frac{y^2}{9} = 1\). From this equation, it can be determined that a^2 = 9 and b^2 = 4.

Step 2: Determine the values of a and b

Since a^2 = 9 and b^2 = 4, the values of a and b can be found by taking the square root of these values. Therefore, a = 3 and b = 2.

Step 3: Determine the position of the foci

The distance from the center (0,0) to each foci is given by the formula \(\sqrt{a^2 - b^2}\). In this case, \(\sqrt{9 - 4} = \sqrt{5}\). So, the positions of the foci are (-\(\sqrt{5}\),0) and (\(\sqrt{5}\),0).

Step 4: Graph the ellipse

Begin by plotting the center of the ellipse at the origin (0,0). Then, draw the major axis from (-3,0) to (3,0) and the minor axis from (0,-2) to (0,2). Sketch the ellipse around these axes so it passes through the end points of the major and minor axes. Lastly, mark the positions of the foci (-\(\sqrt{5}\),0) and (\(\sqrt{5}\),0).

Key Concepts

Graphing Ellipses

Graphing an ellipse involves plotting an oval shape on a coordinate plane. To begin, we identify the center of the ellipse, which in this case is at the origin (0,0). We then need to determine the lengths of the major and minor axes. The major axis is the longest diameter, while the minor axis is the shortest.

To graph:

• Locate the endpoints of the major axis based on the value of 'a' (half-length of the major axis), which is 3 in this example. Plot points at (-3, 0) and (3, 0).
• Determine the endpoints of the minor axis using 'b' (half-length of the minor axis), which is 2. Plot points at (0, -2) and (0, 2).
• Draw a smooth, oval shape that touches all four points, ensuring it is wider along the major axis and shorter across the minor axis.

This oval path is your ellipse, centered at the origin.

Foci of an Ellipse

The foci of an ellipse are two special points located along the major axis. These points help define the shape of the ellipse. Every point on the ellipse is equidistant from the sum of the distances to the two foci.

In the equation, we find the foci using the formula \( \sqrt{a^2 - b^2} \) to calculate the distance from the center to each focus. With our values:

• \( a^2 = 9 \) and \( b^2 = 4 \)
• The distance is \sqrt{9 - 4} = \sqrt{5} \approx 2.24
• The foci are thus located at (-\sqrt{5}, 0) and (\sqrt{5}, 0)

These points lie on the major axis and are essential for understanding the ellipse's symmetry and geometry.

Standard Form of an Ellipse Equation

To properly graph an ellipse and find its foci, you'll need to convert its equation to the standard form. The standard form of an ellipse is:\(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)

This form clearly shows the relationship between the coefficients and the lengths of the axes. For the equation given, \( 9x^2 + 4y^2 = 36 \), we divide each term by 36 to simplify:\(\frac{x^2}{4} + \frac{y^2}{9} = 1\)

Recognizing this formula allows us to pinpoint the values of \( a^2 \) and \( b^2 \), indicating the squares of the semi-axes' lengths. This transformation is crucial for finding the geometric properties of the ellipse.

Major and Minor Axes

The major and minor axes of an ellipse are fundamental for determining its shape and dimensions.

The major axis is the longest diameter running through the center and connecting two endpoints, spanning 2a. Meanwhile, the minor axis is the shortest, measuring 2b.

• The major axis is horizontal in this example, with endpoints at (-3, 0) and (3, 0). This reflects that \( a = 3 \).
• The minor axis is vertical, showing endpoints at (0, -2) and (0, 2), corresponding to \( b = 2 \).

The distinction between major and minor axes helps describe whether the ellipse is elongated horizontally or vertically, guiding how it is graphed on a coordinate plane.