Problem 25 - College Algebra

Question

Find the standard form of the equation of each ellipse satisfying the given conditions. $$\text { Foci: }(-5,0),(5,0) ; \text { vertices: }(-8,0),(8,0)$$

Step-by-Step Solution

Answer

The equation of the ellipse in standard form is \(\dfrac{x^2}{64} + \dfrac{y^2}{39} = 1\)

Step 1: Identify the center, a, b, and c

The center of the ellipse is halfway between the vertices or the foci. This is at the origin (0,0). The length of the major axis is twice the distance from the center to a vertex, so \(2a = 16\), hence \(a = 8\). Similarly, the length of the focal distance is twice the distance from the center to a focus, thus \(2c = 10\), and \(c = 5\).

Step 2: Calculate b

We can now calculate the length of the semi-minor axis, b. By using the relationship between a, b and c in an ellipse, which is \(a^2 = b^2 + c^2\), and plugging in the values a = 8 and c = 5 obtained from step 1, we get \(b = \sqrt{a^2 - c^2} = \sqrt{64 - 25} = \sqrt{39}\).

Step 3: Write Equation of Ellipse in Standard form

The equation of an ellipse with center at the origin (0,0) is given by \(\dfrac{x^2}{a^2} + \dfrac{(y^2}{b^2} = 1\). Replace the lengths a and b in the formula with the values calculated in steps 1 and 2, respectively. So, we get \(\dfrac{x^2}{8^2} + \dfrac{(y^2}{\sqrt{39}^2} = 1\), which simplifies to \(\dfrac{x^2}{64} + \dfrac{y^2}{39} = 1\).

Key Concepts

Standard Form

The standard form of an ellipse's equation is fundamental in understanding its geometry. An ellipse can be represented by an equation in the standard form, which is particularly useful for identifying its key components. For an ellipse centered at the origin, or point (0,0), the standard form of the equation is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Here, \(a\) and \(b\) are, respectively, the semi-major and semi-minor axes of the ellipse. The values of \(a\) and \(b\) are crucial because they set the dimensions of the ellipse. When \(a > b\), the ellipse extends more along the x-axis than the y-axis, forming a horizontally oriented ellipse. Conversely, if \(b > a\), the ellipse is elongated along the y-axis. Understanding how to rearrange the given parameters into the standard form is a significant step in analyzing ellipses.

Foci

The foci of an ellipse are two distinct points located along its major axis. These points hold a unique property: for all points on the ellipse, the sum of the distances to each focus is constant. This balance essentially defines the shape of the ellipse, offering insight into its structure. Identifying the foci can sometimes be more challenging, but they are crucial for understanding the geometric properties of the ellipse. For an ellipse centered at the origin

• The coordinates of the foci are \(( \pm c, 0)\) when the major axis lies along the x-axis.


• The relationship \(c^2 = a^2 - b^2\) assists in finding the foci by connecting them to the semi-major and semi-minor axes.

In the given problem, the foci were (-5,0) and (5,0) which means \(c = 5\). These determined the positioning of the foci along the horizontal axis, which also implied that the ellipse was horizontally stretched.

Vertices

Vertices are key points located on the major axis of an ellipse, which help define its overall structure. The vertices have the maximum and minimum lengths from the center of the ellipse along the major axis. For an ellipse centered at the origin

• The vertices are positioned at \(( \pm a, 0)\) if the ellipse is stretched more along the x-axis.


• The distance between the vertices determines the length of the major axis, given by \(2a\).

In this problem, the vertices were (−8,0) and (8,0), which told us that \(a = 8\), given the total distance between the vertices was 16 units long. Knowing the location of the vertices allowed for assessing the orientation and size of the ellipse efficiently.

Major Axis

The major axis is the longest line segment through the center of an ellipse, spanning from one vertex to the other, and encompassing both foci. This axis is central to understanding the shape's symmetry and scale. Key attributes of the major axis include:

• It defines the primary orientation of the ellipse—horizontal if along the x-axis, and vertical if along the y-axis.


• Its length is twice the length of the semi-major axis, calculated as \(2a\).

In our case, with the given vertices at (−8,0) and (8,0), the major axis stretched horizontally along the x-axis, reflecting the greater horizontal stretch. The presence of the foci (-5,0) and (5,0) along this axis further reinforced this orientation. Recognizing the major axis is a cornerstone in applying the standard form and subsequently grasping the ellipse's geometric configuration.