Problem 86 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Answer: The given equation represents a horizontal ellipse, centered at the origin (0, 0), with a semi-major axis of length 1 along the x-axis and a semi-minor axis of length 1/2 along the y-axis.

Step 1: Identify the Type of Surface

The surface is defined by the equation $$x^2 + 4y^2 = 1.$$ Since there is no z component and the sum of x and y terms involves squares, the equation represents an ellipsoid.

Step 2: Recognize an Ellipse

If we set \(z = 0\) (the constant plane), the equation becomes $$x^2 + 4y^2 = 1$$ This equation represents an ellipse in the \(xy\)-plane, with the center at the origin \((0, 0)\).

Step 3: Determine the Axis Lengths

In general, an ellipse has the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). By comparing the given equation to the general equation, we can identify the values of \(a^2\) and \(b^2\). In our case, \(a^2 = 1\) and \(b^2 = \frac{1}{4}\). Therefore, the semi-major and semi-minor axis lengths are: \(a = 1\) and \(b = \frac{1}{2}\).

Step 4: Describe the Surface

The surface defined by the equation $$x^2 + 4y^2 = 1$$ is a horizontal ellipse, centered at the origin \((0, 0)\), with a semi-major axis of length 1 along the x-axis and a semi-minor axis of length \(\frac{1}{2}\) along the y-axis.