Problem 24 - Precalculus

Question

Find the standard form of the equation of the ellipse with the given characteristics. Foci: (0,0),(0,8)\(;\) major axis of length 16

Step-by-Step Solution

Answer

The standard form of the equation for the ellipse is \(x^2/48 + (y-4)^2/64 = 1\).

Step 1: Determine the Center

Because the foci are at (0,0) and (0,8), the center of the ellipse is the midpoint of these two points. Use the midpoint formula \((x_1 + x_2) / 2, (y_1 + y_2) / 2\). Accordingly, the center is (0,4)

Step 2: Determine \(a\), \(b\), and \(c\)

The length of the major axis is 16, so the semi-major axis (\(a\)) is \(16/2 = 8\). From the given foci we know that the distance from the center to a foci (\(c\)) is 4. The length of the minor axis (\(b\)) can be found using the relationship \(a^2 - c^2 = b^2\). With the obtained \(a\) and \(c\), this calculation gives us \(b^2 = 64 - 16 = 48\)

Step 3: Write the Equation of the Ellipse

The standard form of an ellipse with its center at (0,4) and its major axis vertical is \[(x-h)^2/b^2 + (y-k)^2/a^2 = 1\]. Substituting (0,4) for (h,k), 48 for \(b^2\), and 64 for \(a^2\) gives us our equation \(x^2/48 + (y-4)^2/64 = 1\]

Key Concepts

Foci of an Ellipse

In the world of ellipses, foci (singular: focus) are two fixed points that define the shape. The sum of the distances from any point on the ellipse to these two foci is constant. For the given problem, the foci are positioned at (0,0) and (0,8). This tells us that the major axis is oriented vertically. Another important point is the relationship between foci and the center of the ellipse. Here, the center is the midpoint between the foci, calculated as follows:

• The x-coordinate is (0+0)/2 = 0.
• The y-coordinate is (0+8)/2 = 4.

Thus, the center of the ellipse is at (0,4), playing a crucial role in determining the standard form equation.

Major and Minor Axis

Ellipses have two main axes: the major axis and the minor axis. The major axis is the longest diameter, passing through the center and both foci, while the minor axis is the shorter diameter, perpendicular to the major axis.

• In this problem, the length of the major axis is given as 16.
• The semi-major axis is half of this length, which gives us a value of 8 for the semi-major axis, represented as \(a\).

To find the minor axis representation, we use the formula for ellipses: \(a^2 - c^2 = b^2\). Here, \(c\) is the distance from the center to a focus point, given as 4. Substituting the known values, we find:

• \(b^2 = 8^2 - 4^2 = 64 - 16 = 48\)

So, the ellipse is defined by semi-major axis \(a = 8\) and a minor axis based on \(b^2 = 48\).

Standard Form of an Ellipse

The standard form of an ellipse equation is essential for identifying its properties. With a vertical major axis, the formula looks like this:

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

Here, \((h, k)\) is the center of the ellipse. From the problem, it is known that the center is at (0,4), \(b^2 = 48\), and \(a^2 = 64\).Substituting these values, the standard form equation becomes:

• \(\frac{x^2}{48} + \frac{(y-4)^2}{64} = 1\)

This representation fully embodies the ellipse's geometry, based on its axes and foci.