Question

Explain why the following is true: "If the coefficient of the \(x^{2}\) term in the equation of an ellipse in standard form is smaller than the coefficient of the \(y^{2}\) term, then the ellipse has a horizontal major axis."

Short answer

The smaller coefficient for \(x^2\) implies a longer horizontal axis (\(a > b\)).

Step-by-step solution

Understanding the Standard Form of an Ellipse

The standard equation for an ellipse is either of the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) or \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\). The values \(a\) and \(b\) represent the semi-major and semi-minor axes, respectively.

Identifying the Coefficients

In the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), \(\frac{1}{a^2}\) is the coefficient of the \(x^2\) term and \(\frac{1}{b^2}\) is the coefficient of the \(y^2\) term. Thus, if \(\frac{1}{a^2} < \frac{1}{b^2}\), then \(a^2 > b^2\).

Analyzing the Inequality

Given \(a^2 > b^2\) in the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), it means \(a\) is greater than \(b\). This implies the major axis of the ellipse is along the \(x\)-axis.

Conclusion on Major Axis Direction

Thus, if the coefficient of \(x^2\) (\(\frac{1}{a^2}\)) is smaller than the coefficient of \(y^2\) (\(\frac{1}{b^2}\)), the ellipse must have a horizontal major axis where the length along the \(x\)-axis (\(a\)) is longer than the length along the \(y\)-axis (\(b\)).

Key concepts

Major Axis

The major axis of an ellipse is the longest diameter that runs through the center and both foci. It's the line segment that stretches from one end of the ellipse to the other through its widest part. The major axis defines the overall shape and orientation of the ellipse. If this major axis is vertical, the ellipse appears taller than it is wide; conversely, if it's horizontal, the ellipse is wider than it is tall.

A quick way to remember how the major axis works is to look at the values of the semi-major and semi-minor axes, often denoted as \(a\) and \(b\) respectively. The semi-major axis is half of the major axis and the semi-minor axis is half of the minor axis. In equations where \(a > b\), the major axis is aligned along the respective axis—horizontal for \(x\)-axis and vertical for \(y\)-axis.
In essence:

• A horizontal major axis has longer length along the \(x\)-direction.
• A vertical major axis has longer length along the \(y\)-direction.

It's essential to identify the major axis as it significantly influences the ellipse's orientation and geometry.

Ellipse Equation

The equation of an ellipse is crucial in understanding its shape and properties. In its standard form, an ellipse can be expressed as either \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) or \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\). These equations define the ellipse depending on the relative lengths of its axes.

Key components in the equation include:

• \(a\) and \(b\): These are positive numbers representing the lengths of the semi-major and semi-minor axes, respectively.
• \(a\) is typically greater than \(b\), ensuring the major axis is correctly assigned.

Depending on the equation used, the ellipse will be oriented differently. In \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), the major axis lies along the \(x\)-axis, indicating a more horizontal appearance. Alternatively, \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\) corresponds to a vertical orientation. By carefully analyzing these coefficients and terms, you can easily determine the ellipse's specific attributes.

Coefficient Comparison

In ellipse equations, coefficients help determine the orientation and dimensions of the shape. When comparing coefficients like \(\frac{1}{a^2}\) and \(\frac{1}{b^2}\), you can infer much about the ellipse's characteristics.

Here's how it works:

• If \(\frac{1}{a^2} < \frac{1}{b^2}\), it implies that \(a^2 > b^2\). This means the semi-major axis \(a\) is longer, making the major axis horizontal.
• Conversely, if \(\frac{1}{a^2} > \frac{1}{b^2}\), then \(b^2 > a^2\), positioning the major axis vertically.

The relationship of these coefficients to each other tells us where the ellipse "stretches" more, whether along the \(x\)-axis or \(y\)-axis. This direct comparison allows us to conclude the ellipse's orientation quickly. Thus, understanding this comparison is vital for solving problems involving ellipses efficiently.