Problem 21 - Precalculus: Mathematics for Calculus

Question

An equation of an ellipse is given. (a) Find the vertices, foci, and eccentricity of the ellipse. (b) Determine the lengths of the major and minor axes. (c) Sketch a graph of the ellipse. $$3 x^{2}+y^{2}=9$$

Step-by-Step Solution

Answer

Vertices: (0, ±3), (±√3, 0); Foci: (0, ±√6); Eccentricity: √6/3; Major axis: 6; Minor axis: 2√3.

Step 1: Write the Standard Form

First, we need to rewrite the given equation of the ellipse, \(3x^2 + y^2 = 9\), in its standard form. Divide every term by 9 to get the canonical form: \(\frac{x^2}{3} + \frac{y^2}{9} = 1\). Now it's clear that the major axis is along the y-axis and the minor along the x-axis.

Step 2: Identify the Coefficients

In the standard form, \(\frac{x^2}{3} + \frac{y^2}{9} = 1\), compare it with \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\). We determine that \(a^2 = 9\) and \(b^2 = 3\), so \(a = 3\) and \(b = \sqrt{3}\).

Step 3: Find the Vertices

The vertices along the major axis (here the y-axis) are at \((0, \pm a)\), which equates to \((0, \pm 3)\). The vertices along the minor axis (x-axis) are at \((\pm b, 0)\), or \((\pm \sqrt{3}, 0)\).

Step 4: Calculate the Foci

To find the foci, use the formula \(c = \sqrt{a^2 - b^2}\). Thus, \(c = \sqrt{9 - 3} = \sqrt{6}\). The foci are located at \((0, \pm c)\), which are \((0, \pm \sqrt{6})\).

Step 5: Calculate the Eccentricity

Eccentricity \(e\) is given by \(e = \frac{c}{a}\). We previously found \(c = \sqrt{6}\) and \(a = 3\), so \(e = \frac{\sqrt{6}}{3}\).

Step 6: Lengths of Axes

The length of the major axis is \(2a = 6\) and the length of the minor axis is \(2b = 2\sqrt{3}\).

Step 7: Sketch the Graph

First, draw the axes. Plot vertices at \((0, \pm 3)\) on the y-axis and \((\pm \sqrt{3}, 0)\) on the x-axis. Indicate the foci at \((0, \pm \sqrt{6})\). Draw the ellipse, passing through these points smoothly, ensuring it is elongated along the y-axis due to the major axis.

Key Concepts

Vertices

In an ellipse, the vertices are significant points found at the ends of the major and minor axes. For our ellipse, described by the standard form equation \(\frac{x^2}{3} + \frac{y^2}{9} = 1\), the major axis is aligned with the y-axis.

• To find the vertices on the major axis (y-axis), check where it intersects, resulting as \((0, \pm 3)\).
• For the minor axis (x-axis), the vertices are positioned at \((\pm \sqrt{3}, 0)\).

These points effectively demonstrate the ellipse's reach along its principal dimensions.
Remember, placing vertices correctly is essential in shaping the orientation of an ellipse.

Foci

The foci are crucial in determining the inherent 'stretch' of the ellipse. These internal focal points help portray how the ellipse deviates from being a perfect circle. The foci are always located along the major axis.

• Calculate by using the formula \(c = \sqrt{a^2 - b^2}\).
• For our specific ellipse equation, \(c = \sqrt{9 - 3} = \sqrt{6}\).
• Thus, the foci are positioned at \((0, \pm \sqrt{6})\).

It's noteworthy that these points hover between the center and the vertices along the major axis.

Eccentricity

Eccentricity provides insight into the shape of the ellipse, defining how much it occupies its axes and conveying its deviation from a circular form. The eccentricity \(e\) is calculated with the ratio \(e = \frac{c}{a}\).

• For this ellipse equation, previously calculated \(c = \sqrt{6}\), and \(a = 3\).
• Thus, \(e = \frac{\sqrt{6}}{3}\).

An eccentricity varies between 0 and 1. Avalanches towards 0 reflect a nearly circular shape, whereas values pulling towards 1 signify a more elongated form, which characterizes our ellipse.

Major Axis

The major axis is the longest segment running through the center of the ellipse, indicating its main dimension. In our ellipse, this axis is aligned with the y-axis.

• The length of the major axis is determined as \(2a\).
• Since \(a = 3\), the total length comes to \(2 \times 3 = 6\).
• The vertices at the endpoints of this axis are \((0, \pm 3)\).

This represents the primary stretch of the ellipse, being more significant than that of the minor axis.

Minor Axis

Contrary to the major axis, the minor axis is the shortest segment passing through the center and perpendicular to the major one. For our ellipse, it aligns with the x-axis.

• The minor axis length is given by \(2b\).
• With \(b = \sqrt{3}\), the length equals \(2\sqrt{3}\).
• Vertices on this line rest at \((\pm \sqrt{3}, 0)\).

This axis provides balance and helps define the overall ellipse proportionality.