Problem 4 - Algebra 2

Question

Write an equation of an ellipse with the given characteristics. Check your answers. center \((3,-6),\) vertical major axis of length \(14,\) minor axis of length 6

Step-by-Step Solution

Answer

The equation of the ellipse having center at (3,-6), vertical major axis of length 14, minor axis of length 6 is: \((x-3)^2 / 3^2 + (y+6)^2 / 7^2 = 1\).

Step 1: Identify the Given Parameters

From the provided description, we can identify the center coordinates \((h, k) = (3,-6)\). The length of the vertical major axis is 14, and hence \(a = 14/2 = 7\). The length of minor axis is given as 6 so \(b = 6/2 = 3\).

Step 2: Substitute into Formula

Now, substitute these parameters into the ellipse equation. The vertical axis is longer, so our \(a\) term will correspond with the \(y\) variable in the standard form. This results in the following equation: \((x-3)^2 / 3^2 + (y+6)^2 / 7^2 = 1\).

Key Concepts

Vertical Major Axis

In an ellipse, the major axis is the longest diameter that runs through its center, while the minor axis is the shortest. When we say that an ellipse has a vertical major axis, it means this longest diameter is aligned vertically. This orientation affects how we construct the equation of the ellipse.
In the case of a vertical major axis, the formula emphasizes the variable associated with the vertical direction, usually the y-variable in the Cartesian coordinate system.

• For a vertical major axis, the length is represented by the term involving the y-variable.
• The semi-major axis length is half the total major axis length, denoted as "a".

Knowing whether the major axis is vertical or horizontal is essential for writing the ellipse equation correctly.

Center of an Ellipse

The center of an ellipse serves as a reference point for constructing its equation. It's crucial in conic sections to have an accurate center, as it helps in defining the symmetry. The center of the ellipse is generally denoted by \((h, k)\).

• The values "h" and "k" represent the horizontal and vertical coordinates, respectively.
• This center point shows where the axes intersect.

For our specific exercise, the center is given as (3, -6). This information is vital in positioning the ellipse accurately on a coordinate plane, impacting how both the major and minor axes are drawn.

Conic Sections

Conic sections are the curves obtained by intersecting a cone with a plane. Depending on the angle and position of the intersection, you can get various shapes: circles, ellipses, parabolas, or hyperbolas.
An ellipse arises when the plane cuts through the cone at an angle, avoiding parallelism with the base. Compared to other conic sections, ellipses have some unique characteristics:

• They are symmetric about their axes.
• The sum of distances from any point on the ellipse to its two foci is constant.

Understanding ellipses within the broader context of conic sections allows for comprehension of their properties and equations.

Standard Form of Ellipse

The standard form of the ellipse equation is vital for identifying its attributes and orientation. An ellipse equation comes in two standard forms, aligned around its major axis:

• For a horizontal major axis: \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \).
• For a vertical major axis: \( \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \).

In our exercise, we deal with a vertical major axis. Therefore, the formula used was \( \frac{(x-3)^2}{3^2} + \frac{(y+6)^2}{7^2} = 1 \). Knowing these formulas and how to substitute the respective parameters is key to solving ellipse-related problems.