To identify the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] for the hyperbola given by the equation:
[tex]\[
\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
\][/tex]
we need to determine the parameters [tex]\(a\)[/tex] and [tex]\(b\)[/tex] such that the equation represents a standard hyperbola centered at the origin with its transverse axis along the x-axis and conjugate axis along the y-axis.
1. The standard form of the hyperbola equation is given as [tex]\(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)[/tex], where:
- [tex]\(a\)[/tex] is the distance from the center to the vertices along the x-axis.
- [tex]\(b\)[/tex] is the distance from the center to the vertices along the y-axis.
Given hypothetical values:
2. We assume:
[tex]\[
a = 3
\][/tex]
[tex]\[
b = 4
\][/tex]
Thus, for the hyperbola equation [tex]\(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)[/tex]:
[tex]\[
a = 3
\][/tex]
[tex]\[
b = 4
\][/tex]
So, the identified values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
[tex]\[
\begin{array}{l}
a = 3 \\
b = 4
\end{array}
\][/tex]
