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Magazine Chapter 12: Problem 38
Reduce the equation to one of the standard forms, classify the surface, and sketch it. $$ 4 x^{2}+y^{2}+z^{2}-24 x-8 y+4 z+55=0 $$
Short Answer
Expert verified
The surface is an ellipsoid centered at (3, 4, -2).
Step by step solution
01
Rearrange Terms
First, we'll start by rearranging the given equation to group the terms with the same variables together: \[ 4x^2 - 24x + y^2 - 8y + z^2 + 4z + 55 = 0. \] Observe how the terms with \(x\), \(y\), and \(z\) are grouped for simplicity.
02
Complete the Square for x-terms
For the terms involving \(x\), complete the square. Start with \(4x^2 - 24x\): \[ 4(x^2 - 6x). \] To complete the square, add and subtract \(9\) inside the parentheses: \[ 4((x^2 - 6x + 9) - 9) = 4((x - 3)^2 - 9) = 4(x - 3)^2 - 36. \] This simplifies the \(x\) terms to \(4(x - 3)^2 - 36.\)
03
Complete the Square for y-terms
Now, complete the square for the \(y\) terms: \(y^2 - 8y\). Add and subtract \(16\): \[ (y^2 - 8y + 16 - 16) = ((y - 4)^2 - 16). \] This simplifies the \(y\) terms to \((y - 4)^2 - 16.\)
04
Complete the Square for z-terms
Now, complete the square for the \(z\) terms: \(z^2 + 4z\). Add and subtract \(4\): \[ (z^2 + 4z + 4 - 4) = ((z + 2)^2 - 4). \] This simplifies the \(z\) terms to \((z + 2)^2 - 4.\)
05
Substitute Back and Simplify
Substitute the completed squares back in the equation: \[ 4(x - 3)^2 - 36 + (y - 4)^2 - 16 + (z + 2)^2 - 4 + 55 = 0. \] Simplify the constants: \[-36 - 16 - 4 + 55 = -1. \] Therefore, the equation becomes: \[ 4(x - 3)^2 + (y - 4)^2 + (z + 2)^2 = 1.\]
06
Identify the Surface Type
The equation is now in the standard form of an ellipsoid: \[ \frac{(x-3)^2}{0.25} + \frac{(y-4)^2}{1} + \frac{(z+2)^2}{1} = 1. \] The surface is classified as an ellipsoid centered at \((3, 4, -2)\).
07
Sketch the Ellipsoid
To sketch the ellipsoid, note its center at \((3, 4, -2)\). It extends along the \(x\)-axis to \(0.5\) units (since \(\frac{1}{\sqrt{0.25}} = 2\) making its width 1), along \(y\)-axis and \(z\)-axis to 1 unit. Draw the ellipsoid with the center and these dimensions.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Completing the Square
Completing the square is a technique used to transform a quadratic equation into a perfect square trinomial. This is essential when we want to convert a polynomial into a form that is easy to analyze or graph. In this exercise, completing the square helps to transform the given equation into a recognizable geometric form. Let's break it down:
- For the term related to \(x\), we look at \(4x^2 - 24x\). We factor out the 4, then deal with the expression inside the parenthesis: \(x^2 - 6x\).
- To complete the square, find a number that makes \(x^2 - 6x + ?\) a perfect square. By taking half of \(-6\), which is \(-3\), and squaring it, we get \(9\).
- Add and subtract this number inside the squared term: \((x^2 - 6x + 9) - 9\), leading to \((x - 3)^2 - 9\).
- Reinsert the expression back, multiplying by any factors you pulled out.
Standard Form Equation
Transforming an equation into its standard form is key for classifying a surface. The equation of an ellipsoid is typically presented in standard form as:\[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} + \frac{(z-l)^2}{c^2} = 1\]where \((h, k, l)\) is the center of the ellipsoid and \(a, b, c\) are the semi-axes lengths along the \(x\), \(y\), and \(z\) directions, respectively.
- Through completing the square, our equation \(4(x - 3)^2 + (y - 4)^2 + (z + 2)^2 = 1\) takes a similar form, indicating an ellipsoid.
- Rewriting provides divisions by \(a^2\), \(b^2\), and \(c^2\), necessary for the standard form. For example, dividing through by the coefficient of the squared term shows \(a^2 = (1/4), b^2 = 1, c^2 = 1\).
- The center of the ellipsoid is easily identified from these transformations, which is \((3, 4, -2)\).
Surface Sketching
After deriving the standard form equation of a geometric surface, sketching it provides a visual understanding. A clear sketch can help interpret the dimensions and orientation of the surface.
- First, identify the center of the ellipsoid from the form, which in this case is \((3, 4, -2)\).
- Next, observe the lengths of the axes. Here, they are \(0.5\) units along the \(x\)-axis (since \(a = 0.5\)), and 1 unit along both the \(y\) and \(z\) axes.
- This means the ellipsoid is extended differently in each direction, impacting how it appears. Along \(x\), it is narrower while more rounded along \(y\) and \(z\).
