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Chapter 6: Problem 71

Determine whether the statement is true or false. Justify your answer. If \(D \neq 0\) and \(E \neq 0,\) then the graph of \(x^{2}-y^{2}+D x+E y=0\) is a hyperbola.

Short Answer

Expert verified
The statement is false. Even though the given equation features x and y in both squared and non-squared terms, it doesn't match the structure of a hyperbola's equation.

Step by step solution

01

Identify the given equation

The given equation, \(x^{2}-y^{2}+D x+E y=0\), needs rearranging for comparison purposes.
02

Rearrange the equation

The equation can be rearranged as \(x^{2}- Dx + y^{2} - Ey =0 \). This format allows better comparison to the equation of a hyperbola.
03

Compare with a hyperbola's equation

Considering the structure of a typical hyperbola, \( \frac{(x-h)^{2}}{a^{2}} - \frac{(y-k)^{2}}{b^{2}} = 1 \), the given equation doesn't have the denominator terms which are characteristic for a hyperbola. Therefore, even though it features varying x and y terms, it doesn't conform to a hyperbola's equation.

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