To find the equation of the hyperbola with the given conditions, we need to follow these steps:
1. Identify the parameters:
- Center of the hyperbola: \(\((h, k) = (0, 0)\)\)
- Length of the conjugate axis (\(\(2b\)\)) = 6
- Eccentricity (\(\(e\)\)) = 2
2. Determine \(\(b\)\):
The length of the conjugate axis is given by \(\(2b\)\). So, we can find \(\(b\)\) as:
\(\[
b = \frac{6}{2} = 3
\]\)
3. Square of \(\(b\)\):
Using the value of \(\(b\)\), compute \(\(b^2\)\):
\(\[
b^2 = 3^2 = 9
\]\)
4. Relation between \(\(a\)\), \(\(b\)\), and \(\(e\)\):
The relationship between the semi-major axis (\(\(a\)\)), semi-minor axis (\(\(b\)\)), and the eccentricity (\(\(e\)\)) of a hyperbola is given by:
\(\[
e = \sqrt{1 + \frac{b^2}{a^2}}
\]\)
5. Solve for \(\(a^2\)\):
Given that \(\(e = 2\)\), substitute \(\(e\)\) and \(\(b^2\)\) into the equation:
\(\[
2 = \sqrt{1 + \frac{9}{a^2}}
\]\)
Square both sides to eliminate the square root:
\(\[
4 = 1 + \frac{9}{a^2}
\]\)
Subtract 1 from both sides:
\(\[
3 = \frac{9}{a^2}
\]\)
Multiply both sides by \(\(a^2\)\):
\(\[
3a^2 = 9
\]\)
Divide both sides by 3:
\(\[
a^2 = 3
\]\)
6. Equation of the hyperbola:
Given that the center of the hyperbola is at \(\((0, 0)\)\) and the conjugate axis is along the x-axis, the standard form of a hyperbola with a horizontal transverse axis is:
\(\[
\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
\]\)
Substitute \(\(a^2\)\) and \(\(b^2\)\) into the equation:
\(\[
\frac{x^2}{3} - \frac{y^2}{9} = 1
\]\)
Thus, the equation of the hyperbola is:
\(\[
\boxed{\frac{x^2}{3} - \frac{y^2}{9} = 1}
\]\)
