Sure, let's find the equation of the hyperbola step by step.
1. Identifying key values:
- Since the vertices of the hyperbola are given as [tex]\((\pm \sqrt{10}, 0)\)[/tex], the distance from the center to each vertex is [tex]\(\sqrt{10}\)[/tex]. This gives us the value of [tex]\(a\)[/tex]:
[tex]\[
a = \sqrt{10}
\][/tex]
- The length of the conjugate axis is 14, so the distance from the center to the endpoints of the conjugate axis is half of this length:
[tex]\[
b = \frac{14}{2} = 7
\][/tex]
2. Calculating [tex]\(a^2\)[/tex] and [tex]\(b^2\)[/tex]:
- The value of [tex]\(a^2\)[/tex]:
[tex]\[
a^2 = (\sqrt{10})^2 = 10
\][/tex]
- The value of [tex]\(b^2\)[/tex]:
[tex]\[
b^2 = 7^2 = 49
\][/tex]
3. Formulating the equation:
For a hyperbola centered at the origin with a horizontal transverse axis, the standard form of the equation is:
[tex]\[
\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
\][/tex]
Substituting the values of [tex]\(a^2\)[/tex] and [tex]\(b^2\)[/tex]:
[tex]\[
\frac{x^2}{10} - \frac{y^2}{49} = 1
\][/tex]
4. The final equation:
The equation of the hyperbola is:
[tex]\[
\frac{x^2}{10} - \frac{y^2}{49} = 1
\][/tex]
