Problem 3 - 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. $$ (3,0),(0,-1) $$

Step-by-Step Solution

Answer

The equation of the ellipse is \(x^2/9 + y^2 = 1\).

Step 1: Identify the values of 'a' and 'b'

The location of the vertex (3,0) gives us 'a', the distance from the origin to a vertex along the x-axis. Here, 'a' = 3. Similarly, from the co-vertex coordinates (0,-1), 'b', the distance from the origin to the vertex along the y-axis can be found as 'b' = 1.

Step 2: Plug the values into the standard form

Now that we know the values of 'a' and 'b', we can put them in the standard equation of the ellipse, \(x^2/a^2 + y^2/b^2 = 1\). We get the equation \(x^2/3^2 + y^2/1^2 = 1\) which simplifies to \(x^2/9 + y^2 = 1\).

Key Concepts

Standard Form

The standard form is a type of equation widely used to describe various geometrical shapes like ellipses. An ellipse in the standard form with its center at the origin is expressed as \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). This formula is pivotal for graphing an ellipse, as it summarizes the relationship between the shape's axes and their lengths.
For an ellipse aligned with the coordinate axis and centered at the origin:

• 'a' represents the semi-major axis length. It is the distance from the center to a vertex on the x-axis.
• 'b' represents the semi-minor axis length. It is the distance from the center to a vertex on the y-axis.

The standard form helps in easily identifying and plotting the ellipse by indicating its size and orientation relative to the axes.

Ellipse Equation

The ellipse equation encapsulates a fundamental mathematical relationship to define an ellipse. In mathematics, it combines the variables along with the axes lengths to form a specific relationship. For an ellipse centered at the origin, the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) describes its geometrical shape.
This equation has some key components:

• \(x\) and \(y\) which are the coordinates of any point on the ellipse.
• \(a\) which is the semi-major axis along the x-axis in this problem.
• \(b\) which is the semi-minor axis along the y-axis in this problem.

By substituting the values of 'a' and 'b' derived from the vertices and co-vertices, we can find the specific equation of an ellipse. For the provided example, plugging in our values gives us \( \frac{x^2}{9} + y^2 = 1 \).

Conic Sections

Ellipses are a part of what mathematicians call "conic sections," which are curves obtained by slicing a cone. There are different types of conic sections including circles, ellipses, parabolas, and hyperbolas. Each of these shapes has distinct properties and equations.
An ellipse results when a plane cuts through a cone at an angle that is not perpendicular to its base. It is essentially a squashed or elongated circle. It's important to know:

• Ellipses have two axes of symmetry.
• They possess two focal points or foci.
• The sum of distances from any point on the ellipse to the two foci is constant.

Understanding ellipses as part of the conic sections helps in comparing their properties with other related geometric figures.

Ellipse Vertices

Vertices of an ellipse are crucial for understanding its dimensions and orientation. The vertices are specific points on the ellipse that help define its structure.
For an ellipse centered at the origin:

• The vertices on the major axis are \((\pm a, 0)\) if the major axis is horizontal.
• The vertices on the minor axis are \((0, \pm b)\) if the minor axis is vertical.

Given a vertex at (3, 0) and a co-vertex at (0, -1), this tells us that the horizontal axis is the major one with a length of 6, and the vertical axis is the minor one with a length of 2. These points make it possible to construct or visualize the spatial layout of an ellipse efficiently. By working through these elements, students can master the intricacies of analyzing and plotting ellipses.