Problem 17 - Algebra 2

Question

Find an equation of an ellipse for each given height and width. Assume that the center of the ellipse is \((0,0) .\) $$ h=5 \mathrm{m}, w=2 \mathrm{m} $$

Step-by-Step Solution

Answer

The equation of the ellipse is \(\frac{x^{2}}{(2.5)^{2}} + \frac{y^{2}}{1^{2}} = 1\).

Step 1: Identify the values of a and b

The semi-major axis length (a) is half of the given height and the semi-minor axis length (b) is half of the given width. So, a = h/2 and b = w/2. For this ellipse, a = 5/2 = 2.5 m and b = 2/2 = 1 m.

Step 2: Substitute the values of a and b into the equation of the ellipse

Plug the values of a and b into the general equation of the ellipse \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\). Replacing a with 2.5 and b with 1 yields: \(\frac{x^{2}}{(2.5)^{2}} + \frac{y^{2}}{1^{2}} = 1\).

Key Concepts

Semi-Major Axis

The semi-major axis is a critical component of an ellipse, representing its longest radius. It is essential in defining an ellipse's shape and size. For any ellipse centered at the origin, where the semi-major axis lies along the y-axis, it extends vertically from the center to the top or bottom of the ellipse. Consequently, it determines the height of the ellipse. If the semi-major axis runs along the x-axis, then it decides the width. Here's how to find the length of the semi-major axis:

• Identify the full height or width of the ellipse.
• The semi-major axis is half of this measure.

For example, with a given ellipse height of 5 meters in the problem, the semi-major axis would be half of that, or 2.5 meters. Such calculations help in accurately sketching and understanding the ellipse's overall dimensions.

Semi-Minor Axis

The semi-minor axis is the shortest radius of an ellipse and runs perpendicular to the semi-major axis. It complements the semi-major axis to complete the elliptical shape. If the semi-major axis is vertical, the semi-minor axis stretches across horizontally, reflecting the ellipse's width at its narrowest point. Alternatively, if the semi-major is horizontal, the semi-minor is vertical. Here's how to find the length of the semi-minor axis:

• Determine the ellipse's total width or height that is perpendicular to the semi-major axis.
• The semi-minor axis is then half of this measure.

Following the exercise's dimensions, where the width is 2 meters, the semi-minor axis comes out to be 1 meter. This aspect balances the ellipse's width versus its height, giving it its characteristic oval shape.

Coordinate Geometry

Coordinate geometry, or analytic geometry, involves plotting points, lines, and curves in a coordinate plane to solve geometrical problems. It effectively combines algebra and geometry using the Cartesian coordinate system, which maps equations into geometric shapes.Ellipses in coordinate geometry have specific equations derived from their geometric properties. The standard equation for an ellipse centered at the origin is:\[ \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \]Here, \(a\) and \(b\) represent the lengths of the semi-major and semi-minor axes, respectively. Using coordinate geometry helps describe the size and placement of the ellipse precisely, which assists in various mathematical applications such as physics and engineering.

Center of Ellipse

The center of an ellipse is its midpoint, from which both the semi-major and semi-minor axes extend. It acts as the symmetry point for the ellipse, meaning it is equidistant from all extreme points of the ellipse on either axis. In the given problem, the center of the ellipse is at the origin:

• The point (0, 0) in the Cartesian plane.
• Every ellipse equation can shift depending on its center, but for this exercise, it remains at (0,0).

Having the center at the origin simplifies the equation to just involve the axes’ rotations, without extra shifts in the plane. Such positioning not only eases calculations but also aids in clearly visualizing the shape within a defined coordinate system.