Problem 7 - 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. $$ (-9,0),(0,-2) $$

Step-by-Step Solution

Answer

The equation of the ellipse is \(x^2/81 + y^2/4 = 1\).

Step 1: Find the values of a and b

From the vertex \((-9,0)\), the semi-major axis \(a\) can be obtained by taking the absolute value of the x-coordinate of the vertex, so we have \(a = 9\). From the co-vertex \((0,-2)\), the semi-minor axis \(b\) can be obtained by taking the absolute value of the y-coordinate of the co-vertex, so we have \(b = 2\).

Step 2: Write down the equation of the ellipse

Substituting \(a = 9\) and \(b = 2\) into the standard form equation, we get \(x^2/9^2 + y^2/2^2 = 1\).

Key Concepts

Semi-Major Axis

The semi-major axis of an ellipse represents half of the longest diameter across the ellipse. It is an essential defining feature of the shape. The line segment runs from the center of the ellipse to one of its vertices on the wider side.
The length of the semi-major axis is denoted by the letter \( a \). It determines the extent of the ellipse along its major direction.
To find the value of \( a \) when given a vertex, calculate the absolute value of its coordinate. For example, if the vertex is at (-9, 0), the semi-major axis \( a \) is given by \( | -9 | = 9 \).

• This means the ellipse extends 9 units from the origin on the x-axis.
• This value helps construct the equation for an ellipse.

Semi-Minor Axis

The semi-minor axis is the segment that runs from the center of the ellipse to one of its co-vertices. It is aligned in the shorter direction of the ellipse, distinguishing it from the major axis.
This axis is denoted with the letter \( b \), and crucially affects the shape of the ellipse.
We find \( b \) by examining the co-vertex. For an ellipse centered at the origin, simply take the absolute value of its coordinate. With a co-vertex at (0, -2), we find \( b = | -2 | = 2 \).

• In this case, the semi-minor axis measures 2 units along the y-axis.
• It's shorter than the semi-major axis, defining the narrower side of the ellipse.

Ellipse in Standard Form

An ellipse equation in standard form makes it easy to identify the semi-major and semi-minor axes. This standard equation takes the form:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
where \( a \) and \( b \) are the lengths of the semi-major and semi-minor axes, respectively.

• Use this form to readily discern the dimensions and orientation of the ellipse.
• For example, substituting \( a = 9 \) and \( b = 2 \) gives us: \[ \frac{x^2}{9^2} + \frac{y^2}{2^2} = 1 \]

This equation clearly shows the elliptic relationship and is a foundational step in understanding and graphing ellipses.

Vertices and Co-Vertices

Vertices and co-vertices define key points on the perimeter of an ellipse. The vertices lie along the longer axis (semi-major), while the co-vertices lie on the shorter axis (semi-minor).
The distances from the center of the ellipse to these points help in determining and visualizing the ellipse's shape.

• Vertices: These are located at \( (\pm a, 0) \) for a horizontally directed ellipse.
• Co-Vertices: These are found at \( (0, \pm b) \) for the same alignment.

With the given vertices \((-9, 0)\) and co-vertices \((0, -2)\), these points outline the ellipse as extended more in the x-direction. This direct relationship between these points and the axes they define is fundamental to constructing the standard ellipse equation.