Problem 28 - Precalculus with Limits

Question

In Exercises 19-28, find the standard form of the equation of the ellipse with the given characteristics. Vertices: \((5, 0) (5, 12); \quad\) endpoints of the minor axis: \((1, 6) (9, 6)\)

Step-by-Step Solution

Answer

The standard form of the equation of the ellipse is \((x - 5)^2/16 + (y - 6)^2/36 = 1\).

Step 1: Find the center (h, k)

The center of the ellipse is the midpoint of the vertices or the minor axis. Since both are the same in this case, it can be found using the midpoint formula, which is the average of the x-coordinates and y-coordinates of the endpoints. Here, the center will be the midpoint of either the vertices \((5, 0)\) and \((5, 12)\), or the minor axis \((1, 6)\) and \((9, 6)\). After calculation, the center (h, k) is \((5, 6)\).

Step 2: Determine the length of semi-major axis (a) and semi-minor axis (b)

The semi-major axis length 'a' is the distance from the center to either vertex on the major axis. Subtraction of the y-coordinates of one of the vertices and the center gives the length of a as \(a = 6\). The semi-minor axis length 'b' is the distance from the center to either endpoint on the minor axis. Subtraction of the x-coordinates of one of the endpoints of the minor axis and the center gives the length of b as \(b = 4\).

Step 3: Write the standard form of the ellipse's equation

The standard form of the equation of an ellipse with its major axis vertical is given by: \((x - h)^2/b^2 + (y - k)^2/a^2 = 1\). Substituting the values of h, k, a, and b determined in steps 1 and 2 gives: \((x - 5)^2/4^2 + (y - 6)^2/6^2 = 1\) which can be simplified to \((x - 5)^2/16 + (y - 6)^2/36 = 1\).

Key Concepts

Standard Form

The standard form of the equation for an ellipse is a key concept in understanding its geometric properties. An ellipse can be represented in a specific form, depending on the orientation of its axes.
For ellipses with a horizontal major axis, the standard form is:

• \[\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\]

Meanwhile, when the major axis is vertical, the roles of \(a\) and \(b\) are reversed:

• \[\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1\]

In these equations:

• \((h, k)\) represents the center of the ellipse.
• \(a\) is the semi-major axis length.
• \(b\) is the semi-minor axis length.

Using these formulas allows us to understand and graph the ellipse easily. When solving exercises like finding the equation of an ellipse, substituting known values for these parameters is the key step to write its equation in standard form.

Semi-Major Axis

The semi-major axis in an ellipse is one of the most critical geometric measurements to understand. It is symbolized by \(a\) and represents half of the longest diameter of the ellipse.
If the major axis is vertical, then this measurement will be in the vertical direction, extending from the center to the top or bottom of the ellipse.

• Calculation: Given vertices \((5, 0)\) and \((5, 12)\), calculate \(a\) by finding the distance from the ellipse’s center (at \((h, k) = (5, 6)\)) to one vertex. Therefore, \(a = |12 - 6| = 6\).

The semi-major axis helps define how "stretched" the ellipse is in one direction. A larger \(a\) indicates a more elongated ellipse.
The length of the semi-major axis is a fundamental part when crafting the ellipse's equation in standard form, as it signifies the dominant axis along which the ellipse is oriented.

Semi-Minor Axis

Alongside the semi-major axis, the semi-minor axis is essential for fully describing an ellipse. Known as \(b\), it is half of the shortest diameter.
For a vertical major axis, the semi-minor axis spans horizontally.

• Calculation: Using the endpoints of the minor axis \((1, 6)\) and \((9, 6)\), determine the semi-minor axis similarly by measuring the distance from the center \((5, 6)\) to one of the endpoints, yielding \(b = |9 - 5| = 4\).

A shorter \(b\) suggests a "narrower" ellipse, providing contrast to the curve’s shape defined by \(a\).
In the configuration of an ellipse’s equation in standard form, \(b\) defines the breadth of the ellipse and aids in showing its proportionality.