Problem 9 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of the given equation, indicating vertices, foci, and asymptotes (if it is a hyperbola). $$ \frac{x^{2}}{16}+\frac{y^{2}}{4}=1 $$

Step-by-Step Solution

Answer

It's an ellipse with vertices (±4,0) and (0,±2), and foci at (±2√3,0). No asymptotes.

Step 1: Identify the Type of Conic Section

The given equation is \( \frac{x^{2}}{16} + \frac{y^{2}}{4} = 1 \). This is in the form \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \), which represents an ellipse, not a hyperbola.

Step 2: Determine the Vertices

For the ellipse \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \), the vertices are located at \( (\pm a, 0) \) and \( (0, \pm b) \). Here, \( a = \sqrt{16} = 4 \) and \( b = \sqrt{4} = 2 \). Thus, the vertices are \( (\pm 4, 0) \) and \( (0, \pm 2) \).

Step 3: Find the Foci

The foci of an ellipse are determined using the formula \( c = \sqrt{a^{2} - b^{2}} \). Calculating, \( c = \sqrt{16 - 4} = \sqrt{12} = 2\sqrt{3} \). So, the foci are at \( (\pm 2\sqrt{3}, 0) \).

Step 4: Note there are No Asymptotes

Asymptotes are features related to hyperbolas, not ellipses. Since the given equation represents an ellipse, we do not have asymptotes.

Step 5: Sketch the Ellipse

Draw a coordinate plane. Plot the vertices at \( (4, 0) \), \( (-4, 0) \), \( (0, 2) \), and \( (0, -2) \). Locate the foci at approximately \( (\pm 3.46, 0) \). Sketch the ellipse ensuring it crosses through these vertices and surrounds the foci.

Key Concepts

Conic Sections

Conic sections are curves that arise from the intersection of a right circular cone with a plane. These curves are categorized into four main types: circles, ellipses, parabolas, and hyperbolas. Each conic section has its distinct set of properties and characteristics.

• Circle: A special type of ellipse where the distances from the center to any point on the curve are equal.
• Ellipse: Formed when the plane cuts through the cone at an angle, but doesn’t intersect the base. It resembles an elongated circle.
• Parabola: Occurs when the plane is parallel to the edge of the cone.
• Hyperbola: Results when the plane intersects both halves of the cone.

Conic sections are widely used in various fields such as physics, engineering, and astronomy, owing to their unique geometric properties. Understanding these forms helps in grasping advanced mathematical and real-world concepts.

Vertices of an Ellipse

The vertices of an ellipse are the farthest points on the ellipse along its major and minor axes. These provide valuable information about the size and orientation of the ellipse. For an ellipse centered at the origin with the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), the vertices can be calculated as follows:

• Major Axis: It is the longest axis of the ellipse. The vertices here are at \((\pm a, 0)\).
• Minor Axis: It is the shortest axis of the ellipse. The vertices here are at \((0, \pm b)\).

In our example, \(a = 4\) and \(b = 2\), meaning the vertices are located at \((\pm 4, 0)\) for the major axis and \((0, \pm 2)\) for the minor axis. These points are crucial when sketching the ellipse as they define its overall shape and size.

Foci of an Ellipse

The foci (plural of focus) of an ellipse are two internal points from which the sum of the distances to any point on the ellipse remains constant. This unique property helps in understanding and applying the ellipse's geometry to practical problems. The foci are located along the major axis of the ellipse, and their distance from the center is given by the formula \(c = \sqrt{a^2 - b^2}\).
For the given ellipse, \(a = 4\) and \(b = 2\), meaning:

• First, calculate \(c\): \(c = \sqrt{16 - 4} = \sqrt{12}\).
• The distance \(c\) simplifies to \(2\sqrt{3}\), approximately 3.46.
• Thus, the foci are located at \((\pm 2\sqrt{3}, 0)\). These points help in accurately drawing the elliptical shape, ensuring it correctly encompasses the key properties.

Equation of an Ellipse

The general equation of an ellipse, when centered at the origin in a Cartesian coordinate system, can be expressed as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). In this form, \(a\) and \(b\) denote the lengths of the semi-major and semi-minor axes, respectively. The structure of this equation reveals important characteristics of the ellipse:

• Axis Lengths: \(2a\) is the total length of the major axis, and \(2b\) is the length of the minor axis.
• Orientation: If \(a > b\), the major axis is horizontal; if \(b > a\), it is vertical.
• Symmetry: The equation highlights the ellipse's symmetric properties about both the x and y-axes.

Understanding and applying the equation of an ellipse is fundamental for solving problems related to its geometry. For the given problem \(\frac{x^2}{16} + \frac{y^2}{4} = 1\), the values of \(a = 4\) and \(b = 2\) lead to the correct identification and interpretation of vertices and foci, essential for sketching and analyzing the ellipse.