Problem 47 - College Algebra

Question

For the following exercises, use the given information about the graph of each ellipse to determine its equation. Center at the origin, symmetric with respect to the \(x-\) and \(y\) -axes, focus at \((0,-2)\) , and point on graph \((5,0)\) .

Step-by-Step Solution

Answer

The equation of the ellipse is \( \frac{x^2}{25} + \frac{y^2}{29} = 1 \).

Step 1: Identify the Form of the Ellipse Equation

An ellipse centered at the origin with axes aligned with the coordinate axes will have an equation of the form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a \) and \( b \) are the semi-axis lengths, and \( a > b \) if it's a horizontal ellipse, or \( b > a \) if it's a vertical ellipse.

Step 2: Determine if the Ellipse is Horizontal or Vertical

Given that a focus is at \((0,-2)\), the major axis is along the \(y\)-axis, making this a vertical ellipse. Hence, \( b > a \), where \(b\) is on the \(y\)-axis.

Step 3: Determine the Relationship between \(c\), \(a\), and \(b\)

For a vertical ellipse, the relationship is \( c^2 = b^2 - a^2 \). Given that a focus at \((0,-2)\) implies \( c = 2 \).

Step 4: Use a Point on the Graph to Find \(a\)

The point \((5,0)\) lies on the ellipse. Substitute \(x = 5\) and \(y = 0\) into the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) to find \(a\). This gives \(\frac{25}{a^2} = 1\), so \(a^2 = 25\), therefore, \(a = 5\).

Step 5: Solve for \(b\) Using the Relationship of \(c^2 = b^2 - a^2\)

Substitute \( a^2 = 25 \) and \( c = 2\) into \( c^2 = b^2 - a^2 \). Since \(4 = b^2 - 25\), solving gives \(b^2 = 29\).

Step 6: Write the Equation of the Ellipse

Substitute \( a^2 = 25 \) and \( b^2 = 29 \) into the ellipse equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). This gives the equation \( \frac{x^2}{25} + \frac{y^2}{29} = 1 \).

Key Concepts

Conic Sections

Conic sections refer to the curves formed by intersecting a plane with a double-napped cone. These intersections give rise to four primary types of curves: ellipses, parabolas, hyperbolas, and circles. Each of these shapes has unique properties and equations that describe them.

Ellipses, in particular, are closed curves that represent extended circles. They have some distinguishing features: they possess two axes of symmetry and two foci, which are central to their structure. These characteristics make ellipses important in various fields, such as astronomy, physics, and engineering.

When working with the equations of ellipses, especially those centered at the origin, we use the standard form equation:

• Horizontal ellipse: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) where \( a > b \).
• Vertical ellipse: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) where \( b > a \).

Understanding these distinctions helps in solving more intricate problems related to ellipses.

Foci of Ellipses

The foci (plural of 'focus') are two special points located along the major axis of an ellipse. The property of ellipses is that the sum of distances from any point on the ellipse to the two foci is constant. This property defines the shape and placement of the ellipse.

The distance from the center to each focus is denoted as \( c \), and its relationship with the semi-axis lengths \( a \) and \( b \) is governed by the equation:

• For horizontal or vertical ellipses: \( c^2 = b^2 - a^2 \) (for vertical ellipses when \( b > a \)).

In our exercise, the focus is given at \((0,-2)\), indicating a vertical ellipse with the major axis along the \( y \)-axis.

Understanding the foci is crucial because it helps determine the nature and properties of the ellipse, including eccentricity, which measures how "stretched" the ellipse is.

Axes of Symmetry

In geometry, axes of symmetry are lines that divide a shape into two identical mirror-image halves. In ellipses, there are two axes of symmetry: the major axis and the minor axis.

The major axis is the longest diameter that runs through the foci. It defines the "length" of the ellipse, while the minor axis runs perpendicular to the major axis and represents the "width" of the ellipse.

For ellipses centered at the origin:

• A vertical ellipse has the \( y \)-axis as the major axis and the \( x \)-axis as the minor axis.
• A horizontal ellipse has the \( x \)-axis as the major axis and the \( y \)-axis as the minor axis.

In our example, the focus at \((0, -2)\) signifies that the major axis is along the \( y \)-axis. This symmetry is fundamental to comprehending the nature and graphing of ellipses.

Semi-axis Lengths

The semi-axis lengths of an ellipse are half the lengths of the major and minor axes. They are denoted as \( a \) for the semi-major axis and \( b \) for the semi-minor axis.

These lengths are integral in forming the equation of the ellipse, \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). If our ellipse is vertical, as in this exercise, \( b \) (semi-major) is greater than \( a \) (semi-minor).

For our specific ellipse centered at the origin with a point at \((5,0)\) on its graph, we obtained:

• \( a = 5 \), since the point lies exactly \( 5 \) units from the center along the \( x \)-axis.
• To find \( b \), we use \( c^2 = b^2 - a^2 \), where \( c = 2 \), leading to \( b^2 = 29 \).

Accurate calculation of these lengths is crucial as it impacts the whole representation of the ellipse graphically and algebraically.