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Chapter 12: Problem 86

Identify and briefly describe the surfaces defined by the following equations. $$x^{2}+4 y^{2}=1$$

Short Answer

Expert verified
Answer: The given equation represents an ellipse centered at the origin (0,0) with a major axis of length 2 and a minor axis of length 1. The ellipse is stretched along the x-axis, making the x-axis its major axis and the y-axis its minor axis.

Step by step solution

01

Rewrite the given equation in a recognizable form

We are given the equation $$x^{2}+4 y^{2}=1$$ To rewrite this in a recognizable form, divide both sides of the equation by 1, resulting in: $$\frac{x^{2}}{1} + \frac{4y^{2}}{1} = 1$$ This equation can also be written as: $$\frac{x^{2}}{1} + \frac{y^{2}}{\frac{1}{4}} = 1$$
02

Identify the type of surface

Now, we compare this equation to the standard forms of various types of surfaces to determine which one it matches. The given equation is in the form of an ellipse, with the standard equation being: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ where a and b are the lengths of the major and minor axes, respectively. In this case, a = 1 and b = \(\frac{1}{2}\).
03

Describe the surface

We have identified the given equation as representing an ellipse. The ellipse is centered at the origin (0,0) and has a major axis of length 2a = 2 and a minor axis of length 2b = 1. The ellipse is stretched along the x-axis, making the x-axis its major axis and the y-axis its minor axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Surface Identification
Surface identification is all about recognizing and understanding the type of geometric surface described by a given equation. With the surface equation \(x^{2}+4 y^{2}=1\), we begin by determining its form.

This particular equation is recognized as an equation of an ellipse.

By comparing the coefficients of \(x^2\) and \(y^2\), and checking against the standard forms of various geometric shapes such as ellipses, parabolas, and hyperbolas, we can decide the precise surface. This step is crucial because it lays the foundation for accurately interpreting any equation in mathematics to identify the surface it represents.
Equation Transformation
Understanding equation transformation is necessary when dealing with conic sections like ellipses. In our example, the equation \(x^{2}+4 y^{2}=1\) is initially written in a general form.

To identify it as an ellipse, you can divide through by a constant or manipulate the equation to match the standard form of an ellipse, \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
  • This transformation process helps us compare the coordinates and structure with the standard definitions.
  • It clarifies the values of \(a\) and \(b\), which determine the lengths of the axes of the ellipse.
  • In our equation, this transformation gives \(a = 1\) and \(b = \frac{1}{2}\).

Being able to rewrite equations efficiently can make interpreting and working with mathematical properties far more manageable.
Ellipse Properties
Exploring ellipse properties uncovers various interesting facts about their shape and orientation. Ellipses are defined by several characteristics:
  • They are symmetric with respect to both the x and y axes.
  • Their general shape is an elongated circle, characterized by the lengths of two main axes, the major and minor axes.
For the equation \(x^{2}+4 y^{2}=1\), identified as an ellipse:
  • The major axis is parallel to the x-axis, measuring \(2a = 2\).
  • The minor axis, measuring \(2b = 1\), is parallel to the y-axis.
  • This specific orientation affects the ellipse鈥檚 appearance, making it wider along the x-axis.

Understanding these properties allows you to predict how changes in the equation affect the ellipse's orientation and dimensions.

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Most popular questions from this chapter

Suppose \(n\) houses are located at the distinct points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right) .\) A power substation must be located at a point such that the sum of the squares of the distances between the houses and the substation is minimized. a. Find the optimal location of the substation in the case that \(n=3\) and the houses are located at \((0,0),(2,0),\) and (1,1) b. Find the optimal location of the substation in the case that \(n=3\) and the houses are located at distinct points \(\left(x_{1}, y_{1}\right)\) \(\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\) c. Find the optimal location of the substation in the general case of \(n\) houses located at distinct points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots\) \(\left(x_{n}, y_{n}\right)\) d. You might argue that the locations found in parts (a), (b), and (c) are not optimal because they result from minimizing the sum of the squares of the distances, not the sum of the distances themselves. Use the locations in part (a) and write the function that gives the sum of the distances. Note that minimizing this function is much more difficult than in part (a).

A clothing company makes a profit of \(\$ 10\) on its long-sleeved T-shirts and \(\$ 5\) on its short-sleeved T-shirts. Assuming there is a \(\$ 200\) setup cost, the profit on \(\mathrm{T}\) -shirt sales is \(z=10 x+5 y-200,\) where \(x\) is the number of long-sleeved T-shirts sold and \(y\) is the number of short-sleeved T-shirts sold. Assume \(x\) and \(y\) are nonnegative. a. Graph the plane that gives the profit using the window $$ [0,40] \times[0,40] \times[-400,400] $$ b. If \(x=20\) and \(y=10,\) is the profit positive or negative? c. Describe the values of \(x\) and \(y\) for which the company breaks even (for which the profit is zero). Mark this set on your graph.

Let \(h\) be continuous for all real numbers. a. Find \(f_{x}\) and \(f_{y}\) when \(f(x, y)=\int_{x}^{y} h(s) d s\). b. Find \(f_{x}\) and \(f_{y}\) when \(f(x, y)=\int_{1}^{x y} h(s) d s\).

What point on the plane \(x-y+z=2\) is closest to the point (1,1,1)\(?\)

Find the points (if they exist) at which the following planes and curves intersect. $$8 x+y+z=60 ; \quad \mathbf{r}(t)=\left\langle t, t^{2}, 3 t^{2}\right\rangle, \text { for }-\infty
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