Problem 21 - Precalculus with Limits

Question

In Exercises 19-28, find the standard form of the equation of the ellipse with the given characteristics. Vertices: \((0, 2), (8, 2); \quad\) minor axis of length \(2\)

Step-by-Step Solution

Answer

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

Step 1: Determine the coordinates of the center

The center is at the midpoint of the vertices. Given vertices are \((0, 2)\) and \((8, 2)\). To find the center, average the x-coordinates and the y-coordinates respectively. Center, \(h = (0 + 8) / 2 = 4\) and \(k = (2 + 2) / 2 = 2\). As a result, the coordinates of the center are \((4, 2)\).

Step 2: Calculate the lengths of semi-major and semi-minor axes

The length of major axis is equivalent to the distance between the vertices. The given vertices (\(0, 2\) & \(8, 2\)) are such that they have the same y-coordinate but different x-coordinates, so they are along the x-axis. It means the semi-major axis 'a' is along the x-axis, and is half the distance between the two vertices which is half of 8 i.e., \(a = 8 / 2 = 4\).\nThe minor axis length is given as 2, so the semi-minor axis 'b' is half of this i.e., \(b = 2 / 2 = 1\).

Step 3: Substitution into the standard equation

Substitute the values of h, k, a, and b into the standard form \((x-h)^2/a^2 + (y-k)^2/b^2 = 1\), yielding \((x-4)^2/4^2 + (y-2)^2/1^2 = 1\).

Key Concepts

Standard Form of an Ellipse

The standard form of an ellipse's equation depends on its orientation, whether horizontal or vertical. An ellipse can be thought of as a stretched circle, where the stretching defines the direction of its major axis. The generic standard form of an ellipse centered at a point

• For a horizontal major axis, the equation is: \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]Here, - \((h, k)\) are the coordinates of the center.- \(a\) is the length of the semi-major axis.- \(b\) is the length of the semi-minor axis.
• For a vertical major axis, the roles of \(a\) and \(b\) are reversed:\[ \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \]

Knowing the orientation and dimensions of an ellipse helps greatly in writing its equation in the standard form.

Semi-Major Axis

The semi-major axis is one of the principal characteristics of an ellipse. It is half the length of the major axis, the longest diameter passing through the center. Understanding whether the major axis is horizontal or vertical is essential since it dictates the primary direction of the ellipse. In our exercise:

• The major axis is horizontal as the given vertices are \((0, 2)\) and \((8, 2)\), aligning along the x-axis.
• To find the semi-major axis (\(a\)), calculate half the distance between these vertices. Thus, \(a = \frac{8}{2} = 4\).

Knowing \(a\) in the equation \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \] allows us to position the ellipse accurately on a coordinate plane.

Semi-Minor Axis

The semi-minor axis is the other crucial measurement of an ellipse. Perpendicular to the semi-major axis, it is the shortest distance from the center to the ellipse's perimeter. In context:

• The minor axis length is given as 2.
• The semi-minor axis (\(b\)) then is half this length: \(b = \frac{2}{2} = 1\).

In the equation, \(b\) determines how far the ellipse stretches perpendicular to the major axis. Therefore, this parameter significantly shapes the ellipse, influencing the equation form \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]. Thus, knowing \(b\) helps define the ellipse comprehensively.

Vertices of an Ellipse

Vertices are essential points on an ellipse, specifically the endpoints of the major axis. In the context of our exercise:

• The given vertices are \((0, 2)\) and \((8, 2)\), marking the extent of the ellipse along its major axis.
• These points provide insights into the length and orientation of the major axis. Since both vertices lie on the x-axis, the major axis is horizontal.
• The center of the ellipse lies midway between these vertices, which is computed by averaging the coordinates. Thus, the center is at \((4, 2)\).

Knowing the vertices helps derive the standard form of the equation by identifying the major axis and assisting in finding the center of the ellipse.