Problem 5 - Algebra for College Students

Question

Find the vertices, the endpoints of the minor axis, and the foci of the given ellipse, and sketch its graph. See answer section. $$ 9 x^{2}+3 y^{2}=27 $$

Step-by-Step Solution

Answer

Vertices: (0, ±3); Endpoints of minor axis: (±√3, 0); Foci: (0, ±√6).

Step 1: Write the Equation in Standard Form

The given equation of the ellipse is \(9x^2 + 3y^2 = 27\). Start by dividing the whole equation by 27 to get it into the standard form of an ellipse, \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). This yields:\[\frac{x^2}{3} + \frac{y^2}{9} = 1.\]

Step 2: Identify Major and Minor Axes

The standard form equation \(\frac{x^2}{3} + \frac{y^2}{9} = 1\) indicates that the semi-major axis is along the y-axis (since 9 > 3). Thus, \(a^2 = 9\) and \(b^2 = 3\) leading to \(a = 3\) and \(b = \sqrt{3}\).

Step 3: Find the Vertices

The vertices of the ellipse are at the endpoints of the major axis. Since the major axis is along the y-axis, the vertices are \((0, \pm a)\). Given \(a = 3\), the vertices are \((0, 3)\) and \((0, -3)\).

Step 4: Find Endpoints of the Minor Axis

The endpoints of the minor axis are along the x-axis at \((\pm b, 0)\). With \(b = \sqrt{3}\), the endpoints are \( (\sqrt{3}, 0)\) and \((-\sqrt{3}, 0)\).

Step 5: Calculate the Foci

The foci of the ellipse are along the major axis, calculated using \( c^2 = a^2 - b^2 \). Substitute \(a = 3\) and \(b = \sqrt{3}\): \[c^2 = 9 - 3 = 6\quad \Rightarrow \quad c = \sqrt{6}. \]Thus, the foci are at \((0, \pm \sqrt{6})\).

Step 6: Sketch the Ellipse

To sketch the ellipse, plot the center at the origin \((0, 0)\), draw the major axis from \((0, -3)\) to \((0, 3)\), and the minor axis from \((-\sqrt{3}, 0)\) to \((\sqrt{3}, 0)\). Mark the foci at \((0, \sqrt{6})\) and \((0, -\sqrt{6})\) and draw an ellipse passing through these points, ensuring symmetry about both axes.

Key Concepts

Vertices

In any ellipse, the vertices are crucial as they define the orientation and extent of the major axis. They are located at the endpoints of this major axis. For an ellipse that's centered at the origin, the vertices are found symmetrically around the center, either along the x-axis or y-axis, depending on the orientation. In our solved example, the equation \[\frac{x^2}{3} + \frac{y^2}{9} = 1 \] reveals that the major axis is along the y-axis because the term \(\frac{y^2}{9}\) has the greater denominator, indicating a larger semi-axis.

• This results in the vertices being at the points \((0, 3)\) and \((0, -3)\), as \(a = 3\), based on the step-by-step solution.

Finding the vertices is fundamental when thinking about sketching or mapping the shape of the ellipse, as they pinpoint the extremities on the longer axis.

Minor Axis

The minor axis of an ellipse is the shorter axis that lies perpendicular to the major axis. For the ellipse in this exercise, you will recognize it by identifying which of the fractions in the standard form equation\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]has the smaller denominator; this reflects the smaller radius and thus the minor axis.

• In the example given, the minor axis is along the x-axis, determined by \(b = \sqrt{3}\).
• The equation of the minor axis will then be defined by points \((\pm \sqrt{3}, 0)\).

Understanding the minor axis is essential not only for sketching but also for recognizing the balance that an ellipse conveys around its center.

Foci

The foci of an ellipse are two distinct points on the major axis, evenly distributed around the center. Unlike circles, ellipses have these two foci, which are crucial for defining the shape's geometry. The entire figure can be thought of as a set of points for which the sum of distances from any point on the ellipse to the foci is constant.For our given ellipse, we derived the foci using the relation: \[c^2 = a^2 - b^2\] where \(c\) is the distance from the center to each focus.

• Using \(a = 3\) and \(b = \sqrt{3}\), we find \(c = \sqrt{6}\).
• Therefore, the foci are located at \((0, \pm \sqrt{6})\).

Recognizing the foci not only serves a theoretical purpose but also aids in the precise understanding and plotting of the ellipse.

Standard Form

The standard form of an ellipse's equation is fundamental when analyzing its properties. It is presented as:\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]where \(a\) and \(b\) represent the semi-major and semi-minor axes respectively. This expression helps easily determine the ellipse's characteristics like vertices, axes, and foci. In our specific problem, converting the given equation \[9x^2 + 3y^2 = 27\] to standard form involved dividing by 27:\[\frac{x^2}{3} + \frac{y^2}{9} = 1\]

• This step standardizes the ellipse equation and highlights that the major axis is aligned along the y-axis.
• It allows us to definitively calculate other components like the vertices and the foci.

Understanding the standard form is critical to efficiently dissecting the elliptical shape and applying this to problem-solving in geometry.