Problem 32 - A Graphical Approach to Precalculus with Limits

Question

Find an equation for each ellipse. Center \((2,0) ;\) minor axis of length \(6 ;\) major axis horizontal and of length 9

Step-by-Step Solution

Answer

Equation: \(\frac{(x-2)^2}{20.25} + \frac{y^2}{9} = 1\).

Step 1: Identify the Parameters of the Ellipse

The given center is \((2, 0)\). Since the major axis is horizontal, the ellipse has the form \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), where \((h,k)\) is the center of the ellipse. Thus, \(h = 2\) and \(k = 0\).

Step 2: Determine the Lengths of Semi-Major and Semi-Minor Axes

The major axis has a length of 9, which means the semi-major axis \(a = \frac{9}{2} = 4.5\). The minor axis has a length of 6, which means the semi-minor axis \(b = \frac{6}{2} = 3\).

Step 3: Write the Equation of the Ellipse

Substitute \(h = 2\), \(k = 0\), and the values of \(a\) and \(b\) into the standard form equation. The equation becomes: \[\frac{(x-2)^2}{4.5^2} + \frac{(y-0)^2}{3^2} = 1.\] Simplifying, this results in: \[\frac{(x-2)^2}{20.25} + \frac{y^2}{9} = 1.\]

Key Concepts

Standard Form Equation

An ellipse is a shape that looks like a stretched circle. The standard form equation for an ellipse is a specific way of writing down its mathematical formula. This equation helps us understand the size and shape of the ellipse quickly.

For an ellipse centered at a point \((h,k)\), its standard form is \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]. Here's what each part of this equation means:

• \((h, k)\) is the center of the ellipse. It tells us where the ellipse is located on the coordinate plane.
• The terms \((x-h)^2\) and \((y-k)^2\) measure how far a point \((x, y)\) on the ellipse is from its center.
• \(a\) and \(b\) are the semi-major and semi-minor axes, respectively, determining the shape and size of the ellipse.
• The denominator \(a^2\) lies under the \(x\)-part if the major axis is horizontal, and under the \(y\)-part if it's vertical.

By substituting our given values, we get the specific equation for the ellipse, which becomes very useful for understanding and working with the ellipse.

Semi-Major Axis

In an ellipse, the semi-major axis is half the length of the ellipse's longest diameter. This axis outlines the stretching width of the ellipse.

For our given problem:

• The total length of the major axis is 9 units.
• To find the semi-major axis, you divide the length by 2: \( a = \frac{9}{2} = 4.5 \).

The semi-major axis plays a crucial role in forming the standard form equation. It is represented by the variable \(a\), and its square \(a^2\) goes into the equation as the denominator under the \((x-h)^2\) term when the major axis is horizontal.

Understanding the semi-major axis helps describe how far the ellipse extends along its wider direction, giving us a clear idea about its shape.

Semi-Minor Axis

The semi-minor axis of an ellipse is half the length of the shortest diameter. It represents how much the ellipse is compressed along its shorter direction.

In our exercise:

• The total length of the minor axis is provided as 6 units.
• The semi-minor axis is then half of this measurement: \( b = \frac{6}{2} = 3 \).

The semi-minor axis is noted as \(b\), and its square, \(b^2\), is used in the standard form equation beneath the \((y-k)^2\) term.

Grasping the concept of the semi-minor axis aids in understanding how the ellipse shapes vertically or along its less elongated side. It complements the semi-major axis to give a complete picture of the ellipse's proportions in creation of the equation.