Problem 13 - Beginning and Intermediate Algebra

Question

Identify the center of each ellipse and graph the equation. $$\frac{x^{2}}{36}+\frac{y^{2}}{16}=1$$

Step-by-Step Solution

Answer

The center of the ellipse is at (0, 0), with a semi-major axis length of 6 units and a semi-minor axis length of 4 units. To graph the ellipse, plot the center point (0, 0) on the Cartesian plane and move 6 units left/right and 4 units up/down to create a boundary rectangle. Draw the ellipse within this rectangle, ensuring the curve is smooth and symmetric.

Step 1: Identify the center (h, k)

From the given equation, the center is (0, 0) since the equation is already in the form of the general ellipse equation. The center coordinates are (h, k) = (0, 0).

Step 2: Identify the lengths of semi-major and semi-minor axes (a and b)

Again, comparing the given equation with the general ellipse equation, we can see that: \(a^{2} = 36\), and therefore, \(a = \sqrt{36} = 6\) \(b^{2} = 16\), and therefore, \(b = \sqrt{16} = 4\) Thus, the length of the semi-major axis is 6 units, and the length of the semi-minor axis is 4 units.

Step 3: Graph the ellipse

To graph the ellipse, follow these steps: 1. Identify the center point (0, 0) and plot it on the Cartesian plane. 2. From the center, move 6 units to the left and right (as per the semi-major axis, a) and 4 units up and down (as per the semi-minor axis, b). 3. These points will create a boundary around the center, forming a rectangle. The ellipse will be inscribed within this rectangle. 4. Draw the ellipse curve by connecting these points, making sure the curve is smooth and symmetric around the center point. Now, you should have an ellipse with center (0, 0), a length of 12 units along the x-axis (major axis), and a length of 8 units along the y-axis (minor axis).

Key Concepts

Center of an Ellipse

The center of an ellipse is a crucial part of its geometry. For the ellipse represented by the equation \[\frac{x^{2}}{36} + \frac{y^{2}}{16} = 1\], the center is at the origin (0,0). This means that the ellipse is perfectly symmetrical around this central point. In a standard form ellipse equation, centered at (h, k), the equation is \[ \frac{(x - h)^{2}}{a^{2}} + \frac{(y - k)^{2}}{b^{2}} = 1 \]. Here, since the terms \(x^2\) and \(y^2\) lack separate h and k values, it indicates h = 0 and k = 0. Thus, the center is (0, 0).

The center helps in positioning the ellipse on a graph, allowing you to draw it accurately by extending its semi-major and semi-minor axes from this point.

Semi-Major Axis

The semi-major axis (\(a\)) of an ellipse is the longest radius extending from the center to the edge of the ellipse. In the equation \[ \frac{x^{2}}{36} + \frac{y^{2}}{16} = 1 \], the term under \(x^{2}\) is 36. This represents \(a^{2}\), so we find \(a\) by calculating the square root of 36, which is 6.

Therefore, the semi-major axis measures 6 units. A complete major axis, which runs the full length across the ellipse through the center, measures 12 units. You can identify the direction of elongation of the ellipse by the variable in the equation with the larger denominator. Here, since 36 > 16, the ellipse stretches outwards more prominently along the x-axis.

Semi-Minor Axis

The semi-minor axis (\(b\)) is the shortest distance from the center to the edge along the perpendicular direction to the semi-major axis. In the given equation \[ \frac{x^{2}}{36} + \frac{y^{2}}{16} = 1 \], we find \(b\) by taking the square root of 16, yielding a result of 4.

This means the semi-minor axis measures 4 units, with the minor axis totaling 8 units in length along the y-axis. While the semi-major axis defines the longer direction of the ellipse, the semi-minor axis indicates width and shows that the ellipse stretches less along the y-axis.