Problem 84 - Calculus Early Transcendentals

Question

Identify and briefly describe the surfaces defined by the following equations. $$x^{2}+y^{2}+4 z^{2}+2 x=0$$

Step-by-Step Solution

Answer

Answer: The given equation represents an ellipsoid with center at (-1, 0, 0), major axes lengths 1, and minor (vertical) axis length 1/2.

Step 1: Complete the square

To complete the square, group the terms with the same variables together: $$\left(x^2+2x\right)+y^2+4z^2=0$$ Now, complete the square for the x-terms by adding and subtracting the square of half of the coefficient of x: $$\left(x^2+2x+1\right)+y^2+4z^2=1$$ Notice that, since we added \(1\), we must also add it to the other side of the equation.

Step 2: Rewrite the equation

Now, rewrite the equation in a more recognizable form for quadratic surfaces: $$(x+1)^2+y^2+4z^2=1$$

Step 3: Identify the quadratic surface

Compare the equation with the standard equation of an ellipsoid, which is defined as follows: $$\frac{(x-a)^2}{A^2}+ \frac{(y-b)^2}{B^2} + \frac{(z-c)^2}{C^2} = 1$$ Our equation can be written in this form, with center coordinates \((a, b, c) = (-1, 0, 0)\), major radii \(A = 1, B = 1\), and a minor (vertical) radius \(C = \frac{1}{2}\) as: $$\frac{(x+1)^2}{1^2} + \frac{y^2}{1^2} + \frac{z^2}{(\frac{1}{2})^2} = 1$$

Step 4: Describe the Surface

Based on the comparison with the standard equation of an ellipsoid, the given equation $$x^2+y^2+4z^2+2x=0$$ describes an ellipsoid with center at \((-1,0,0)\) and major axes lengths \(1\) and minor (vertical) axis length \(\frac{1}{2}\).