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Chapter 10: Problem 35

How can you tell from the equation of an ellipse whether its graph is horizontal or vertical?

Short Answer

Expert verified
Compare the denominators: \(a^2\) under \(x\) means horizontal, under \(y\) means vertical.

Step by step solution

01

Identify the Standard Form of the Ellipse Equation

An ellipse equation can be written in two standard forms: Horizontal Ellipse: \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\) Vertical Ellipse: \(\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1\)
02

Compare the Denominators

Look at the denominators of the fractions in the equation. For the horizontal ellipse, the term \(a^2\) will be under \((x - h)^2\), and for the vertical ellipse, the term \(a^2\) will be under \((y - k)^2\).
03

Determine the Orientation

If \(a^2\) (the larger denominator) is under \((x - h)^2\), the ellipse is horizontal. If \(a^2\) (the larger denominator) is under \((y - k)^2\), the ellipse is vertical.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

standard form of ellipse equation
To understand if an ellipse is horizontal or vertical, we first need to know the standard form of its equation. An ellipse can be expressed in two different standard forms, depending on its orientation.

The general standard form of an ellipse is:
  • For a horizontal ellipse: \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \)
  • For a vertical ellipse: \(\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1 \)
This standard form helps you identify the ellipse and its key properties, such as the center \((h, k)\) and the lengths of the semi-major and semi-minor axes. Understanding this form is crucial to determining the ellipse's orientation.
horizontal ellipse
A horizontal ellipse is an ellipse where the longer axis (major axis) is along the x-axis. This means the ellipse stretches out more horizontally than vertically.

In the standard form of the equation for a horizontal ellipse \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\), the term \(a^2\) (which represents the square of the semi-major axis) is under \((x - h)^2\).

This placement indicates that the distance the ellipse extends horizontally (the length of the major axis) is greater than the distance it extends vertically (the length of the minor axis). Remember, in a horizontal ellipse: \(a > b\).
vertical ellipse
A vertical ellipse is oriented such that its longer axis (major axis) is along the y-axis. In simpler terms, it extends more vertically than horizontally.

The standard form of the equation for a vertical ellipse is given by \(\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1\). Here, the term \(a^2\) (representing the square of the semi-major axis) is underneath \((y - k)^2\).

This indicates that the distance the ellipse extends vertically is greater than the distance it extends horizontally. In a vertical ellipse, you'll find: \(a > b\).
denominators comparison
To determine whether an ellipse is horizontal or vertical, look closely at the denominators in its equation.

Identify the value of \(a^2\) and \(b^2\). These values indicate the lengths of the axes:

  • For a horizontal ellipse: \(a^2 > b^2\) and \(a^2\) appears under \((x - h)^2\)
  • For a vertical ellipse: \(a^2 > b^2\) and \(a^2\) appears under \((y - k)^2\)

By identifying the larger denominator's placement, you can ascertain whether the ellipse is horizontal or vertical. This simple comparison provides a clear distinction between the two orientations.

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