Problem 16 - Precalculus with Limits

Question

In Exercises 11-18, find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Foci: \((\pm2, 0); \quad\) major axis of length \(10\)

Step-by-Step Solution

Answer

The standard form of the equation of the ellipse with the given characteristics and center at the origin is \(\frac{x^2}{25} + \frac{y^2}{21} = 1\).

Step 1: Determine the Value for Semi-Major Axis length

Firstly, half the length of the major axis is the value of \(a\). In this case, the major axis has a length of \(10\), so the semi-major axis length will be \(a = \frac{10}{2} = 5\).

Step 2: Determine the Value for \(c\) - distance from the center to each focus

The foci are given by the points \((\pm2, 0)\). The distance of focal points from the center represents \(c\). Thus, \(c = 2\).

Step 3: Calculate the Value for Semi-Minor Axis Length

Now you can determine \(b\), the semi-minor axis length. The relationship between the variables \(a\), \(b\), and \(c\) is \(c^2 = a^2 - b^2\). Plugging in the known values, the formula is rearranged to solve for \(b^2\), giving \(b^2 = a^2 - c^2\), and by substituting the known values we get \(b = \sqrt{5^2 - 2^2} = \sqrt{21}\).

Step 4: Write the Standard Form of Ellipse

With the values of \(a\) and \(b\) known, you can substitute these values into the standard form of an ellipse, \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), giving the final equation as \(\frac{x^2}{25} + \frac{y^2}{21} = 1\).

Key Concepts

Standard Form of an Ellipse

The standard form of an ellipse is a way of expressing its equation that makes it easier to understand and visualize. For an ellipse centered at the origin, the standard form is written as follows:

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

Where:

• \(a\) is the length of the semi-major axis.
• \(b\) is the length of the semi-minor axis.

To determine if an ellipse is horizontal or vertical, look at the values of \(a\) and \(b\). If \(a > b\), the ellipse is horizontal, stretching more along the x-axis. If \(b > a\), it is vertical, stretching along the y-axis. This specific formula helps in calculating other important components of the ellipse, such as its axes and foci. By substituting the known values of \(a\) and \(b\), you can express your ellipse in this simplified form.

Semi-Major Axis of an Ellipse

The semi-major axis of an ellipse is one of its most crucial elements. It represents half of the longest diameter of the ellipse. In our example, the total length of the major axis is given as \(10\).

To find the semi-major axis, simply halve this length:

\[ a = \frac{10}{2} = 5 \]

The semi-major axis runs from the center of the ellipse to a point on its longest perimeter, and it's labeled by \(a\).

Understanding the semi-major axis helps determine the ellipse's size and shape. When \(a\) is larger, your ellipse stretches wider along its major direction, defining the ellipse's overall orientation.

Foci of an Ellipse

The foci of an ellipse are two special points located along its major axis. These points are crucial in the geometric definition of an ellipse, as they help dictate the shape and spread of it. For an ellipse centered at the origin, the foci's coordinates are given as \((\pm c, 0)\) when the major axis is horizontal.

In the given problem, the foci are \((\pm2, 0)\). Thus, the distance from the center to each focus is:

\[ c = 2 \]

The foci play an important role in ensuring that the total distance from any point on the ellipse to the two foci is constant. Understanding the foci is essential to grasping how ellipses behave geometrically. They help manage the `bending` of the ellipse beyond the center point.

Semi-Minor Axis of an Ellipse

The semi-minor axis is the half-length of the shortest diameter of an ellipse. Even though it is shorter than the semi-major axis, it is as significant for describing the ellipse's geometry. The semi-minor axis is perpendicular to the major axis and intersects at the ellipse's center.

To find it, recall the formula relating \(a\), \(b\), and \(c\) given by:

\[ c^2 = a^2 - b^2 \]

Plug in the known values to solve for \(b\):

\[ b^2 = 5^2 - 2^2 = 21 \]
\[ b = \sqrt{21} \]

This calculation shows how far the ellipse stretches in its minor direction. Like \(a\), \(b\) defines the ellipse's spread, but it's more about how 'tall' or 'short' the ellipse appears in its perpendicular span.