Problem 23 - A Graphical Approach to College Algebra

Question

Find an equation for each ellipse. \(x\) -intercepts \((\pm 4,0) ;\) foci \((\pm 2,0)\)

Step-by-Step Solution

Answer

The equation is \(\frac{x^2}{16} + \frac{y^2}{12} = 1\).

Step 1: Identify Ellipse Orientation

Since both the x-intercepts and foci are given on the x-axis, the major axis of the ellipse is horizontal.

Step 2: Determine Center and Semi-Major Axis

Since the x-intercepts are \((\pm 4, 0)\), this implies the center is at the origin \((0,0)\) and the semi-major axis length is \(a = 4\).

Step 3: Find Semi-Minor Axis Length

The foci are at \((\pm 2, 0)\), so \(c = 2\). Use the relationship \(c^2 = a^2 - b^2\) to find \(b\). Compute \(b^2 = a^2 - c^2 = 16 - 4 = 12\), so \(b = \sqrt{12} = 2\sqrt{3}\).

Step 4: Write Ellipse Equation

Substitute the found values into the horizontal ellipse equation: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). This gives \(\frac{x^2}{16} + \frac{y^2}{12} = 1\).

Key Concepts

Understanding x-intercepts of an Ellipse

In the context of an ellipse, finding the x-intercepts means identifying the points where the ellipse passes through the x-axis. These are the real roots or solutions to the equation when set to zero for the y-variable. When you plot an ellipse on a coordinate plane, the x-intercepts are the points where the ellipse touches or crosses the x-axis. For the given example, the x-intercepts are \((\pm 4, 0)\). This indicates that the ellipse crosses the x-axis at \(-4\) and \(+4\). This is a characteristic of the horizontal major axis of the ellipse. These x-intercepts help establish the boundary and provide clues about the orientation and size of the ellipse.

The Role of the Foci in an Ellipse

The foci of an ellipse are two distinct points on the major axis, around which the ellipse is shaped. The sum of the distances from any point on the ellipse to the two foci is constant. In the provided example, the foci are located at \((\pm 2, 0)\). These points are crucial because they define the eccentricity of the ellipse, which describes how "stretched" the ellipse appears.

• Foci closer to one another result in a more circular shape.
• Foci that are farther apart produce a more elongated ellipse.

Knowing the foci helps in understanding the geometric nature and symmetry of an ellipse.

Defining the Semi-Major Axis

The semi-major axis of an ellipse is half of the longest diameter that runs through its center and both foci. In our scenario, this is 4 units long, as indicated by the x-intercepts. The semi-major axis is crucial in defining the size of the ellipse. It stretches outward from the center to the edge of the ellipse along the major axis.

• It influences how wide or flat the ellipse appears.
• A longer semi-major axis means a larger ellipse.
• In a formula, it appears as \(a^2\) in the denominator of the term with the x-variable in the standard form of the ellipse's equation.

The semi-major axis also helps identify the overall scale and proportions of the ellipse compared to the semi-minor axis.

Ellipse Orientation in Relation to the Axes

The orientation of an ellipse refers to whether its major axis is aligned horizontally or vertically. For this particular ellipse, since both the x-intercepts and foci are situated on the x-axis, it is classified as having a horizontal orientation. This means:

• The larger diameter (semi-major axis) is aligned along the x-axis.
• The smaller diameter (semi-minor axis) is perpendicular, along the y-axis.

Understanding the orientation is crucial because it affects the standard form of the ellipse's equation, which is different for horizontal and vertical ellipses. In this case, the equation is of the form \(\frac{x^2}{16} + \frac{y^2}{12} = 1\), showing the horizontal stretch along the x-axis. The orientation impacts how we interpret the position and shape of the ellipse on a coordinate plane.