91Ó°ÊÓ

The standard form of the equation for a hyperbola describes its shape and orientation. Depending on whether it's horizontal or vertical, the formula changes slightly. Here, because the y-coordinates of the given vertices and foci are identical, the hyperbola is horizontal. Thus, the equation has the form:

• Horizontal: \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \)
• Vertical: \( \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \)

In both versions,

• \((h,k)\) is the center of the hyperbola
• \(a^2\) and \(b^2\) are squared constants related to the distances of the vertices and co-vertices from the center

Vertices and foci are crucial in defining a hyperbola's shape. The vertices are the closest points on each hyperbola branch to its center. In this specific instance:

• The vertices are at \((-2, 5)\) and \((6, 5)\)
• Thus, they lie along a horizontal line

The foci, on the other hand, are points inside each branch of the hyperbola. They are further from the center than the vertices:

• For our hyperbola, the foci are at \((-3, 5)\) and \((7, 5)\)
• The line joining the foci also confirms the hyperbola's horizontal orientation

The center of a hyperbola is located right in the middle between the vertices. It acts like the balance point of the hyperbola. Here's how you can find it:

• Average the x-coordinates of the vertices: \( h = \frac{-2 + 6}{2} = 2 \)
• Average the y-coordinates of the vertices: \( k = \frac{5 + 5}{2} = 5 \)

So, the center is at \((2, 5)\). This center becomes critical as it allows us to plug \(h\) and \(k\) into our hyperbola's standard equation, correctly positioning it on the graph.

Deriving the equation of a hyperbola involves several orderly steps. Follow these to ensure accuracy:1. **Identify the Hyperbola Type:** - Check the orientation using vertices and foci.2. **Calculate \(a\) and \(a^2\):** - Measure the full distance between vertices: \(2a = 8\) - Thus, \(a = 4\) and \(a^2 = 16\)3. **Calculate \(c\) and \(c^2\):** - Measure the full distance between foci: \(2c = 10\) - So, \(c = 5\) and \(c^2 = 25\)4. **Compute \(b^2\):** - Use the relationship \(c^2 = a^2 + b^2\) - Substitute the known values to find \(b^2 = 9\)5. **Write the Standard Equation:** - Plug all these values \((h=2, k=5, a^2=16, b^2=9)\) into the horizontal standard form: \[ \frac{(x-2)^2}{16} - \frac{(y-5)^2}{9} = 1 \] This structured approach ensures a solid grasp of the hyperbola and its equation.