Problem 4 - 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. $$ (0,6),(1,0) $$

Step-by-Step Solution

Answer

The standard form equation of the ellipse is \(x^2+ \frac{y^2}{36}=1\).

Step 1: Determine the value of 'a'

The value of 'a' corresponds to the distance from the center to a vertex along the major axis. Since the given vertex is (0,6), the distance from the center to the vertex, which is 'a', will be 6.

Step 2: Determine the value of 'b'

The value of 'b' is the distance from the center to a co-vertex along the minor axis. Given that the co-vertex is (1,0), the distance from the origin to the co-vertex, which is 'b', will be 1.

Step 3: Write the equation of the ellipse

Substituting the values of 'a' and 'b' into the standard form of the equation of an ellipse, the equation becomes \(\frac{x^2}{1^2}+\frac{y^2}{6^2}=1\), which simplifies to \(x^2+ \frac{y^2}{36}=1\).

Key Concepts

Ellipse Standard Form

An ellipse is a geometric shape that looks like a stretched circle. To describe it mathematically, we use an equation called the "ellipse standard form." The standard form of an ellipse equation is written as: \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] In this equation, 'a' and 'b' are crucial values that define the ellipse's shape and size. Here is how you can think about the parts:

• \(x\) and \(y\) represent any point on the ellipse.
• \(a\) is the length from the center of the ellipse to a vertex on the horizontal axis, also known as the semi-major axis.
• \(b\) is the length from the center to a co-vertex on the vertical axis, known as the semi-minor axis.
• "1" on the right side of the equation ensures the total shape remains consistent as an ellipse.

Knowing the values of 'a' and 'b' allows us to plot the ellipse's graph accurately, illustrating its size and orientation.

Major and Minor Axes

The major and minor axes are lines that describe the main directions and dimensions of an ellipse. Understanding them is key to working with ellipses effectively.

• Major Axis: The longest line that runs through the center of the ellipse. The vertices lie on this axis, and its total length is equal to twice the value of 'a' (i.e., \(2a\)). For example, in the ellipse given in our exercise, if 'a' is 6, then the major axis is 12 units long.
• Minor Axis: The shortest line through the center of the ellipse. This axis runs perpendicular to the major axis. The co-vertices lie on the minor axis, and its total length is twice the value of 'b' (i.e., \(2b\)). For the exercise, if 'b' is 1, the minor axis is 2 units long.

The axes together help in determining the overall spread of the ellipse in the plane and aide in sketching it accurately.

Vertices and Co-Vertices

The vertices and co-vertices are significant points on the ellipse that help define its shape. Here's how to locate and interpret them:

• Vertices: These are the points of intersection of the ellipse with its major axis. Since the major axis is the longer axis, vertices are the farthest points from the center. In our example, the given vertex is at \((0,6)\). This point specifies that 'a' is 6.
• Co-Vertices: Located at the ends of the minor axis, these points are closer to the center than the vertices. In our exercise, the co-vertex \((1,0)\) indicates that 'b' is 1.

With both vertices and co-vertices identified, you can precisely draw the ellipse. These points also make it easier to understand and use the equation of the ellipse for various calculations or transformations.