Problem 44 - Algebra and Trigonometry

Question

Find an equation for the ellipse that satisfies the given conditions. Eccentricity: \(\sqrt{3} / 2,\) foci on \(y\) -axis, length of major axis: 4

Step-by-Step Solution

Answer

The equation is \( x^2 + \frac{y^2}{4} = 1 \).

Step 1: Identify the ellipse orientation

Since the foci are on the y-axis, the major axis is vertical. Thus, the standard form of the ellipse is \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \) where \( a > b \).

Step 2: Use major axis to find 'a'

The length of the major axis is given as 4. Thus, the semi-major axis \( a \) is half of that: \( a = 2 \).

Step 3: Relate eccentricity to find 'c'

The eccentricity \( e \) is given by \( e = \frac{c}{a} \). We are given \( e = \frac{\sqrt{3}}{2} \). So, \( \frac{c}{2} = \frac{\sqrt{3}}{2} \), which gives \( c = \sqrt{3} \).

Step 4: Use the relationship between a, b, and c

For an ellipse, \( c^2 = a^2 - b^2 \). Substituting \( c = \sqrt{3} \) and \( a = 2 \), we have \( (\sqrt{3})^2 = 2^2 - b^2 \). Therefore, \( 3 = 4 - b^2 \). Solving for \( b^2 \), we find \( b^2 = 1 \).

Step 5: Write the equation of the ellipse

Now that we have \( a^2 = 4 \) and \( b^2 = 1 \), the equation of the ellipse is: \( \frac{x^2}{1} + \frac{y^2}{4} = 1 \), simplifying to \( x^2 + \frac{y^2}{4} = 1 \).

Key Concepts

Eccentricity

Eccentricity is a fundamental concept when it comes to ellipses, describing how much the ellipse is stretched out compared to a perfect circle. An ellipse with an eccentricity of zero is a perfect circle, while one with an eccentricity close to one is more elongated. The eccentricity, denoted as \( e \), defines the shape of the ellipse and is calculated using the formula:

• \( e = \frac{c}{a} \)

where \( c \) is the distance from the center of the ellipse to each focus, and \( a \) is the semi-major axis.In our example, the eccentricity is given as \( \frac{\sqrt{3}}{2} \), which means the ellipse is fairly elongated, but not too stretched. This value helps us determine the internal distances of the ellipse, specifically related to its foci and axes.

Foci

The foci (plural of focus) are two specific points located inside every ellipse. These points play a crucial role in the definition and properties of the ellipse. An ellipse can be defined as the set of all points such that the sum of the distances from two fixed points (the foci) is constant.In a standard vertical ellipse like in our example, the foci lie on the y-axis. The distance \( c \) from the center of the ellipse to each focus is a measure used to position them. Thus, if the center of the ellipse is at the origin (0,0), the coordinates for the foci are:

• (0, \( c \))
• (0, -\( c \))

From our example, calculating using \( c = \sqrt{3} \), the foci are located at (0, \( \sqrt{3} \)) and (0, -\( \sqrt{3} \)). This supports the ellipse's eccentricity and affects the shape of its curve.

Major Axis

The major axis of an ellipse is its longest diameter, passing through the center and both foci. This axis determines the overall length and orientation of the ellipse.For ellipses where the foci are on the y-axis, the major axis is vertical. The length of the major axis is given directly, but it is twice the length of the semi-major axis \( a \), so:

• Length of major axis = \( 2a \)

In the example provided, the length of the major axis is 4, hence the semi-major axis length \( a = 2 \). The major axis aids in understanding the ellipse's orientation and its stretching in the primary direction. The values of \( a \) and \( b \) (semi-minor axis) define the ellipse's equation accurately.

Semi-Major Axis

The semi-major axis is half of the major axis, thus representing the largest radius of the ellipse. It is denoted by \( a \) and is critical in defining the ellipse's dimensions and position.The semi-major axis is used in the primary equation of an ellipse:

• \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \)

where \( a > b \) for a vertical ellipse. In our solved example with a major axis length of 4, the semi-major axis is \( a = 2 \). The semi-major axis length impacts the determination of other related values:

• Helps calculate eccentricity \( e \)
• Links with \( b \) for complete equation of the ellipse