Problem 33 - Algebra and Trigonometry

Question

Find an equation for the ellipse that satisfies the given conditions. Foci \(:( \pm 4,0),\) vertices: \(( \pm 5,0)\)

Step-by-Step Solution

Answer

The equation of the ellipse is \(\frac{x^2}{25} + \frac{y^2}{9} = 1\).

Step 1: Understand the structure of an ellipse equation centered at the origin

The general equation for an ellipse centered at the origin with a horizontal major axis is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \(a\) is the distance from the center to a vertex on the major axis and \(b\) is the distance from the center to a vertex on the minor axis. Here, \(a > b\).

Step 2: Identify given values

The foci are \(( \pm 4, 0)\) which shows that the distance from the center to each focus, \(c\), equals 4. The vertices are \(( \pm 5, 0)\), meaning \(a = 5\).

Step 3: Apply the relation \(c^2 = a^2 - b^2\)

Using the relationship for ellipses \(c^2 = a^2 - b^2\), substitute the known values: \(c = 4\) and \(a = 5\). Therefore, \(16 = 25 - b^2\).

Step 4: Solve for \(b^2\)

Re-arrange the equation to find \(b^2\): \(b^2 = 25 - 16 = 9\).

Step 5: Write the equation of the ellipse

Substitute the values obtained for \(a^2\) and \(b^2\) into the standard form equation: \(\frac{x^2}{25} + \frac{y^2}{9} = 1\). This is the equation of the ellipse.

Key Concepts

Conic Sections

Conic sections are the curves obtained when a right circular cone is intersected by a plane. The four basic types of conic sections are circles, ellipses, parabolas, and hyperbolas. An ellipse is formed when the plane cuts through both halves of the cone at an angle, but is not parallel to the base of the cone or the vertical axis. Ellipses are characterized by their elongated oval shape, which can be oriented either horizontally or vertically. The standard equation for an ellipse can vary depending on this orientation, and the position of the ellipse is categorized by its center and the lengths of its major and minor axes.

Foci of Ellipse

The foci of an ellipse are two fixed points located inside the ellipse. They play a vital role in defining the shape of an ellipse. The unique property of an ellipse is that the sum of the distances from any point on the ellipse to the two foci is constant.For the ellipse centered at the origin and given in the form: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] the foci determine how "stretched" the ellipse is. The distance from the center of the ellipse to each focus is denoted by \(c\), and it relates to the distances \(a\) and \(b\) of the ellipse through the equation: \[ c^2 = a^2 - b^2 \]. This equation helps us find \(c\) when \(a\) and \(b\) are known, as it was in our example where \(c = 4\).

Vertices of Ellipse

Vertices of an ellipse are the points where the ellipse is widest and narrowest. They lie on the major and minor axes of the ellipse. There are two sets of vertices:

• Major Vertices: These are the furthest points along the major axis from the center. In our exercise, they are \( \pm 5, 0 \) since the ellipse is horizontally oriented.
• Minor Vertices: These are the points where the ellipse is narrowest. They lie along the minor axis.

Understanding the location of vertices is crucial because they help define the size and orientation of the ellipse. The fixed distance from the center to these vertices along the axes gives parameter values \(a\) and \(b\) for the ellipse equation. The major axis length, being the longest, is \(2a\) and minor axis length is \(2b\).

Major and Minor Axes

The major and minor axes of an ellipse are the lines that define its breadth and length.

• Major Axis: This is the longer axis of the ellipse. It passes through the foci and both vertices of the ellipse. The length of the major axis is \(2a\).
• Minor Axis: This is the shorter axis, perpendicular to the major axis at the center. Its length is \(2b\), and it helps determine the height of the ellipse.

The relationship between the axes is essential in forming the ellipse equation. Since \(a = 5\) in our exercise, this makes the major axis longer than the minor axis, which measured \(b = 3\) after solving \(c^2 = a^2 - b^2\). The knowledge of these axes helps visualize the shape and orientation of the ellipse as well as calculate its properties efficiently.