Problem 17 - Contemporary Precalculus

Question

Identify the conic section whose equation is given and find its graph. If it is a circle, list its center and radius. If it is an ellipse, list its center, vertices, and foci. $$\frac{y^{2}}{49}+\frac{x^{2}}{81}=1$$

Step-by-Step Solution

Answer

Question: Identify the conic section represented by the equation \(\frac{y^2}{49} + \frac{x^2}{81} = 1\), and find its center, vertices, and foci. Answer: The given equation represents an ellipse, with center at (0,0), vertices at (-9, 0) and (9, 0), and foci at \((-\sqrt{32}, 0)\) and \((\sqrt{32}, 0)\).

Step 1: Identify the conic section

This equation is in the form of an ellipse, which is written as: $$\frac{(x-h)^{2}}{a^2}+\frac{(y-k)^{2}}{b^2}=1$$ Where \((h,k)\) is the center, and \(a\) and \(b\) are the semi-major and semi-minor axes respectively. Our equation is: $$\frac{y^{2}}{49}+\frac{x^{2}}{81}=1$$

Step 2: Find the center

We can match the coefficients in the equations to find the center \((h,k)\) of the ellipse. Since there are no constants added or subtracted from the x and y terms, we know that the center is at the origin (0,0): $$h=0, k=0$$ Center: (0,0)

Step 3: Find the semi-major and semi-minor axes

Again, we can match the coefficients in the equations. Comparing the denominator of the fractions, we have: $$a^2 = 81 \Rightarrow a = 9$$ $$b^2 = 49 \Rightarrow b = 7$$

Step 4: Find the vertices and foci

The vertices are located at a distance of \(a\) units from the center along the major axis. Since the semi-major axis is horizontal, we have to add/subtract 'a' from x-coordinate of the center: Vertices: \((-9, 0)\text{ and }(9, 0)\) To find the foci, we will use the relationship between a, b, and the distance from the center to the foci, c: $$c=\sqrt{a^2-b^2}$$ Plugging in the values for a and b, we get: $$c=\sqrt{81-49}=\sqrt{32}$$ The foci are located at a distance of \(\sqrt{32}\) units from the center, also along the major axis. So, for our ellipse, the foci are: Foci: \((-\sqrt{32}, 0)\text{ and }(\sqrt{32}, 0)\) In conclusion, the given equation represents an ellipse with center (0,0), vertices at (-9, 0) and (9, 0), and foci at \((-\sqrt{32}, 0)\) and \((\sqrt{32}, 0)\).

Key Concepts

Conic Sections

Understanding conic sections is fundamental to grasping how different curves are represented in mathematics, including ellipses, parabolas, and hyperbolas. Conic sections arise when a plane intersects a double cone (imagine two cones that meet at their apexes). The angle and position of the intersecting plane determine the type of conic section formed.

When the plane cuts perpendicularly through the axis of the cone, it forms a circle, a special case of an ellipse. When the plane cuts through the cone at an angle not equal to the angle of the cones, an ellipse is produced. A parabola is formed when the plane is parallel to the slant of the cone. If the plane cuts both nappes (the upper and lower portions of the cone), a hyperbola results.

The study of these conics is not just academic. Conic sections have practical applications in fields such as astronomy, physics, and engineering. For instance, they describe the orbits of planets and satellites, the shapes of certain reflective surfaces like mirrors and lenses, and stress patterns in materials.

Ellipse Graphs

Graphing an ellipse involves understanding its standard form and orientation. The equation of an ellipse generally looks like: \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\]where \((h, k)\) is the center of the ellipse, and \(a\) and \(b\) represent the lengths from the center to the vertices along the major and minor axes, respectively.

Depending on whether \(a > b\) or \(b > a\), the major axis may be horizontal or vertical. When \(a > b\), the ellipse is wider than it is tall and the major axis is horizontal. Conversely, if \(b > a\), the ellipse is taller than it is wide, indicating a vertical major axis.

When graphing the ellipse described in the original exercise, \(\frac{y^2}{49} + \frac{x^2}{81} = 1\), it is crucial to note that since \(\frac{49}{81} < 1\), the major axis is horizontal as 81 is greater than 49—making \(x\) represent the major axis. Knowing the graph's orientation helps in accurately plotting the ellipse on a coordinate plane.

Vertices and Foci of Ellipses

The vertices and foci of an ellipse are key features that help define its shape and orientation. These points provide important information about the size and structure of the ellipse.

- **Vertices**: The vertices of an ellipse are the points where the ellipse is widest or tallest. These are found along the major axis, at a distance equal to \(a\) (the semi-major axis) from the center. For the given ellipse \(\frac{y^2}{49} + \frac{x^2}{81} = 1\), the vertices are at \((-9, 0)\) and \((9, 0)\). This reflects that the major axis stretches horizontally since the larger number (81) is under the \(x^2\) term.

- **Foci**: The foci of an ellipse are points along the major axis, from which the sum of the distances to any point on the ellipse is constant. To find the foci, use the formula \(c = \sqrt{a^2 - b^2}\). For our specific ellipse, the foci are located at \((-\sqrt{32}, 0)\) and \((\sqrt{32}, 0)\). The presence of these foci characterizes the path traced by the ellipse, emphasizing its nature as a set of points maintaining equal overall distance to these two points.

These elements—vertices and foci—not only aid in graphing ellipses but also in understanding their geometric properties and how they relate to real-world phenomena like orbital paths.