Problem 6 - Algebra 2

Question

Write an equation of an ellipse in standard form with center at the origin and with the given vertex and co-vertex. $$ (-6,0),(0,5) $$

Step-by-Step Solution

Answer

The equation of the ellipse in standard form with center at the origin and with the given vertex and co-vertex is \( \frac{x^2}{36} + \frac{y^2}{25} = 1\).

Step 1: Identify Semi-major (a) and Semi-minor(b) Axis

The distance from the center to vertex along the x-axis will be the semi-major axis 'a'. Here it is 6 units (from 0 to -6). Similarly, the distance from the center to the co-vertex along the y-axis will be the semi-minor axis 'b'. Here it is 5 units.

Step 2: Plug in the values into the equation of ellipse centered at origin

The standard formula for ellipse centered at origin is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Based on step 1, we have found that a = 6 and b = 5. So, plug these values into the standard equation, we get \( \frac{x^2}{6^2} + \frac{y^2}{5^2} = 1\)

Key Concepts

Ellipse with Center at Origin

An ellipse is a geometric shape that looks like an elongated circle. When we talk about an ellipse centered at the origin, it simply means the center of the ellipse is at the point \((0,0)\) in the coordinate plane. This is especially useful because it simplifies the equation's form and makes calculations easier. In an ellipse centered at the origin, the axes are symmetric around this center point, offering a straightforward approach to understanding its parameters.

Semi-major Axis

The semi-major axis of an ellipse is one of its most important characteristics. It refers to the longest radius from the center of the ellipse to its edge.

• In the given exercise, the vertex \((-6,0)\) tells us that the semi-major axis stretches along the x-axis by 6 units from the origin.
• This is because the x-coordinate varies between \(-6\) and \(6\), confirming a total major axis length of 12 units, thus making the semi-major axis length 6.

The length of this axis is denoted by \('a'\) in the ellipse equation, and it plays a crucial role in defining the ellipse's shape and size.

Semi-minor Axis

Next, let's discuss the semi-minor axis. It is the shortest radius of the ellipse extending from the center.

• In our specific example, the co-vertex \((0,5)\) indicates the semi-minor axis lies along the y-axis, reaching 5 units from the origin.
• Thus, the total minor axis extends from \(-5\) to \(5\), resulting in a semi-minor axis of length 5.

This axis, denoted by \('b'\) in the ellipse equation, determines how much the ellipse is "squeezed" vertically. Understanding both the semi-major and semi-minor axes allows a complete description of the ellipse's dimensions.

Standard Form of Ellipse

The standard form of an ellipse's equation is a universal way to describe such shapes when centered at the origin. The formula is given by:\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]Here:

• \(a\) is the length of the semi-major axis, defining the shape's stretch along the x-axis.
• \(b\) is the length of the semi-minor axis, indicating the shape's stretch along the y-axis.

Using \(a = 6\) and \(b = 5\) from our exercise, we plug the values into the formula, resulting in:\[\frac{x^2}{6^2} + \frac{y^2}{5^2} = 1\]This equation offers a simple mathematical representation of the ellipse's geometry, helpful for plotting and performing further calculations.