Question

In Exercises \(11-18,\) identify and sketch the quadric surface. Use a computer algebra system to confirm your sketch. $$ 9 x^{2}+y^{2}-9 z^{2}-54 x-4 y-54 z+4=0 $$

Short answer

The given equation represents a hyperboloid of one sheet. The sketch of the surface using a computer algebra system should show a hyperboloid centered at \((3,2,-3)\) with radii 1 and 3, which opens along the z-axis.

Step-by-step solution

Group the Terms

Rearrange and group similar terms together to simplify the equation. Group \(x\), \(y\), and \(z\) terms together: \((9x^2-54x)+(y^2-4y)+(-9z^2-54z)+4=0.\)

Complete the Squares

Use the completing the square method to further simplify the equation. For each of the grouped terms, add and subtract the square of half the coefficient of the linear term. Also, ensure to balance the equation by adding the same terms on the right side of equality. This simplification leads to: \((9(x-3)^2 -81)+(y-2)^2-4+(-9(z+3)^2+81)+4=0.\)

Simplify the Equation into Standard Form

Simplify the equation to arrive at the standard form using the derived results from completing the square method. After this step, the equation becomes: \((x-3)^2+(y-2)^2/9-(z+3)^2=1.\)

Identify the Type of the Quadric Surface

From the standard equation, it can be inferred that the given equation represents a hyperboloid of one sheet because only one term (the \(z\) term) is subtracted in the standard equation. This is the form for a hyperboloid of one sheet: \( \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}-\frac{(z-l)^2}{c^2}=1.\) Here, \( h=3, a=1, k=2, b=3, l=-3, c=1.\)

Sketch and Validate using a Computer Algebra System

Graph the equation using a computer algebra system such as GeoGebra, MatLab, or WolframAlpha for validation. The sketch should show a hyperboloid centered at \((3,2,-3)\) with radii 1 and 3, which opens along the z-axis.