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Chapter 13: Problem 43

Identify the quadric surface. $$ 25 x^{2}+25 y^{2}-z^{2}=5 $$

Short Answer

Expert verified
The identified quadric surface is a hyperboloid of one sheet.

Step by step solution

01

Identify the indicative terms from the equation

The given equation is \(25x^{2}+25y^{2}-z^{2}=5\). In this equation, all coefficients are square numbers, the x and y terms have the same coefficient and the term with variable z has a negative sign.
02

Simplify the equation

Rewrite the equation to resemble the standard forms of quadric surfaces. Divide each side by 5 to get \(5x^{2}+5y^{2}-z^{2}=1\). This further simplifies to \(x^{2}+y^{2}-\frac{1}{5}z^{2}=1\).
03

Identify the quadric surface

Now the equation resembles the standard form of a hyperboloid of one sheet. Its standard form is \(x^{2} + y^{2} - cz^{2} = 1\), where c > 0. In this case, c = 1/5, which is a positive number. Therefore, the quadric surface in the problem is a hyperboloid of one sheet.

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