Problem 45 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the eccentricity of the ellipse. $$\frac{x^{2}}{4}+\frac{y^{2}}{9}=1$$

Step-by-Step Solution

Answer

The eccentricity of the given ellipse is \(e = \frac{\sqrt{5}}{3}\).

Step 1: Identify Semi-major and Semi-minor Axes

From the given equation of the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\), it can be observed that the square of the semi-major axis \(a\) is 9 and the square of the semi-minor axis \(b\) is 4. Therefore, \(a=3\) and \(b=2\).

Step 2: Plug into the Eccentricity Formula

The eccentricity \(e\) is given by \(e = \sqrt{1 - (\frac{b}{a})^2}\). Substituting the values \(a=3\) and \(b=2\) in this formula, we get \(e = \sqrt{1 - (\frac{2}{3})^2}\).

Step 3: Simplify and Solve the Equation

Simplify the equation to calculate the exact value of \(e\). Doing the division inside the brackets first, we get \(e = \sqrt{1 - (\frac{4}{9})}\). Simplifying further, by subtracting \(4/9\) from 1, we get \(e = \sqrt{\frac{5}{9}}\). Calculating the square root of \(5/9\) will give the correct value of the eccentricity \(e\).

Key Concepts

Understanding Semi-Major and Semi-Minor Axes

At the heart of understanding an ellipse is grasping the concepts of the semi-major and semi-minor axes. But what are these axes? In simple terms, imagine an ellipse as an elongated circle. The longest line that can be drawn across it is called the major axis, and as you might guess, the semi-major axis is exactly half of this distance, representing the longest radius of the ellipse. Correspondingly, the shortest line that spans the ellipse is dubbed the minor axis, with the semi-minor axis being half of that length.

In our exercise, the given equation \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\) reveals the lengths of these axes squared. Specifically, the denominator under the \(x^2\) term represents the semi-major axis when it's larger, and the denominator under the \(y^2\) term gives you the semi-minor axis. Here, \(a^2 = 9\) and \(b^2 = 4\), meaning the semi-major axis \(a\) is 3 units, and the semi-minor axis \(b\) is 2 units. Knowing these values is critical as they define the shape and size of the ellipse and are used to calculate various properties, like the eccentricity.

Applying the Eccentricity Formula

Eccentricity is a measure of how 'stretched' an ellipse is. A circle has an eccentricity of 0 because it's perfectly uniform in all directions, but as an ellipse becomes more elongated, its eccentricity approaches 1. To quantify this characteristic, we use the eccentricity formula \(e = \sqrt{1 - (\frac{b}{a})^2}\).

Let's connect this to our exercise. With the values for \(a\) and \(b\) identified, plugging them into the formula is straightforward. We know that \(a = 3\) and \(b = 2\), so we calculate the square of \(\frac{b}{a}\), which equates to \(\frac{2}{3}\). Square this value to get \(\frac{4}{9}\), and then subtract that from 1. This process yields \(e = \sqrt{1 - (\frac{4}{9})}\), which simplifies to \(e = \sqrt{\frac{5}{9}}\). Therefore, the ellipse's eccentricity, a fundamental descriptor of its shape, can be calculated from knowing just the lengths of the axes.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is like tidying up a room – we aim to make everything neat and clear. When simplifying the eccentricity expression, we must be methodical. The expression \(e = \sqrt{1 - (\frac{4}{9})}\) may look a bit daunting, but fear not. First, we perform the subtraction inside the square root. Knowing that \(\frac{9}{9} - \frac{4}{9} = \frac{5}{9}\), we can easily simplify it to \(e = \sqrt{\frac{5}{9}}\).

The expression is much cleaner now, but we're not done yet. The square root of \(\frac{5}{9}\) is the final piece. While it might be tempting to seek a decimal, in mathematics, we often prefer to leave things as simplified radicals or fractions. The square root of a fraction is simply the square root of the numerator over the square root of the denominator, so we end up with \(e = \frac{\sqrt{5}}{3}\), which is a neat expression for the eccentricity of the ellipse from our problem.