Question

Sketch the graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work. $$x^{2}+\frac{y^{2}}{9}=1$$

Short answer

Answer: The coordinates of the vertices are (0, 3) and (0, -3), and the coordinates of the foci are (0, 2.83) and (0, -2.83). The length of the major axis is 6, and the length of the minor axis is 2.

Step-by-step solution

Identify the type of ellipse

The given equation is $$x^2 + \frac{y^2}{9} = 1.$$ Since the y-term is larger than the x-term, this indicates that the ellipse is a vertical ellipse. This means the major axis will be vertical, and the minor axis will be horizontal.

Identify and plot the coordinates of the centre, vertices, and foci

To find the centre of the ellipse, we look for the coordinates (h, k) in the equation. In this case, the equation is already in standard form, so the centre is at the origin (0, 0). Next, we will find the values for a and b (the lengths of the semi-major and semi-minor axes). We can determine that: - a^2 = 9 (the denominator of the y-term) - b^2 = 1 (since there is no coefficient of the x-term) Therefore, a = 3 and b = 1. Now, we will find the vertices of the ellipse. For a vertical ellipse, the vertices are (h, k+a) and (h, k-a). Since the centre is at the origin (0, 0), this yields: - Vertices: (0, 3) and (0, -3) To find the foci, we need to find the distance (c) between the centre and each focus. We can use the relationship c^2 = a^2 - b^2, which yields c^2 = 9 - 1 = 8, and therefore c = sqrt(8) ≈ 2.83. For a vertical ellipse, the foci are (h, k+c) and (h, k-c), which gives: - Foci: (0, 2.83) and (0, -2.83) Now, plot the coordinates of the vertices and foci onto a graph.

Find the lengths of the major and minor axes

We've already found the values for a and b (the lengths of the semi-major and semi-minor axes). To find the lengths of the major and minor axes, simply multiply the semi-axes by 2: - Major axis length: 2 * a = 2 * 3 = 6 - Minor axis length: 2 * b = 2 * 1 = 2 Finally, use a graphing utility to check your work, ensuring that the ellipse has been accurately sketched and labelled.

Key concepts

Conic Sections

In mathematics, conic sections refer to the curves derived by intersecting a plane with a double napped cone. These sections include circles, ellipses, parabolas, and hyperbolas. Each of these curves has unique properties and equations. The equation of our focus here is \( x^2 + \frac{y^2}{9} = 1 \), which represents an ellipse. An ellipse is defined as the set of all points such that the sum of the distances from two fixed points (foci) is constant. This specific conic section showcases the beautiful symmetry and balance found in nature and mathematics.

• Circles can be seen as a special type of ellipse, where the lengths of the axes are equal.
• Ellipses have two different axis lengths, giving them their stretched circular shape.
• Parabolas resemble an open curve, focusing on a single point.
• Hyperbolas consist of two separate curves, diverging away from one another.

By exploring conic sections, one can appreciate the profundity of geometric relationships and their applications.

Graphing Ellipses

Graphing an ellipse involves identifying its key features from its equation. For example, we look at the equation \( x^2 + \frac{y^2}{9} = 1 \). This is a standard form equation of an ellipse centered at the origin \((0,0)\).
First, it's important to determine whether the ellipse is vertical or horizontal. This is determined by comparing the denominators of each term. Here, since the denominator under the \(y\) term is larger, the ellipse is vertical, meaning its major axis is aligned with the \(y\)-axis.Next, we extract key elements:

• Semi-major axis: This is derived from the term with the larger denominator. In this case, \(a^2 = 9\), so \(a = 3\).
• Semi-minor axis: This is from the smaller denominator, \(b^2 = 1\), therefore \(b = 1\).

With \(a\) and \(b\) determined, you can plot the ellipse by starting at the center and marking the vertices and endpoints of the minor axis. This gives a clear visualization of the shape and orientation of the ellipse.

Vertices and Foci

In an ellipse, vertices and foci are crucial for determining its shape and position. They provide insight into the geometric properties of the ellipse. Let's start with the vertices for the given equation \( x^2 + \frac{y^2}{9} = 1 \). Since the ellipse is vertical, the vertices are positioned along the \( y \)-axis and calculated as:

• Vertices: \((0, a)\) and \((0, -a)\). For \(a = 3\), this means the vertices are at \((0, 3)\) and \((0, -3)\).

Next, the foci are points located along the major axis, found using the formula \(c^2 = a^2 - b^2\). Solving \(c^2 = 9 - 1 = 8\) yields \(c ≈ 2.83\). Thus, for a vertical ellipse:

• Foci: \((0, c)\) and \((0, -c)\), so they are at \((0, 2.83)\) and \((0, -2.83)\).

By plotting the vertices and foci, you can more effectively understand how the ellipse is structured, enhancing your graphing accuracy.