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