91Ó°ÊÓ

To understand the equation of a hyperbola, starting with the vertices is crucial. The vertices of a hyperbola are two fixed points that define the shape and orientation of the hyperbola.
The hyperbola given in our problem has vertices at (-3, 0) and (3, 0).
These points indicate that the hyperbola opens horizontally.
The distance between these vertices is called the transverse axis.
In this case, the distance is 6 units.
From the vertices, we can find the value of 'a', which represents half of this distance.
In our example, \(2a = 6\) so \(a = 3\).
Knowing a is essential for forming the equation of the hyperbola.
Thus, the vertices help locate the hyperbola's center and determine its basic structure.

The standard form of the hyperbola's equation depends on its orientation, either horizontal or vertical.
For a hyperbola with a horizontal transverse axis, the equation is:
\(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).
Here, 'a' is the distance from the center to each vertex along the x-axis, and 'b' is related to the distance along the y-axis, crucial for shaping the hyperbola.
When a hyperbola is centered at the origin, as in our problem, the center is (0,0).
The general form simplifies to the standard format by substituting the known values of a and b.
Starting with:\(\frac{x^2}{9} - \frac{y^2}{b^2} = 1\).
This equation format helps identify how the hyperbola stretches and compresses along x and y axes.

To complete the hyperbola's equation, we need to find 'b', the other parameter besides 'a'.
We use a given point on the hyperbola to determine 'b'.
In our example, the hyperbola passes through (-6, -1).
Substituting this point into the standard form equation:
\( \frac{(-6)^2}{9} - \frac{(-1)^2}{b^2} = 1 \).
We calculate it step-by-step:
\( \frac{36}{9} - \frac{1}{b^2} = 1 \)
=> \( 4 - \frac{1}{b^2} = 1 \)
=> \( 3 = \frac{1}{b^2} \)
=> \( b^2 = \frac{1}{3} \).
Thus, we have a necessary parameter to write the final equation.
Plugging 'a^2' and 'b^2' back gives us:
\(\frac{x^2}{9} - \frac{y^2}{\frac{1}{3}} = 1\).
Simplifying, our final hyperbola equation is:
\(\frac{x^2}{9} - 3y^2 = 1\).