the equation of the hyperbola with (3,0) as a focus and (1,0) as a vertice

To find the equation of a hyperbola with a given focus and vertex, we need to follow these steps:

### Step 1: Understand the Hyperbola Structure
A hyperbola centered at \( (h, k) \) with a horizontal transverse axis has the general equation:
\[
\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
\]
- \( (h, k) \) is the center of the hyperbola.
- \( a \) is the distance from the center to each vertex.
- \( c \) is the distance from the center to each focus, with \( c > a \).
- The relationship between \( a \), \( b \), and \( c \) is given by:
\[
c^2 = a^2 + b^2
\]

### Step 2: Identify the Center, Vertex, and Focus
Given:
- The vertex is \( (1, 0) \).
- The focus is \( (3, 0) \).

Since the vertex is at \( (1, 0) \) and the focus is at \( (3, 0) \), the center \( (h, k) \) of the hyperbola is the midpoint between these two points:
\[
h = \frac{1 + 3}{2} = 2, \quad k = 0
\]
So, the center is \( (2, 0) \).

### Step 3: Determine \( a \) and \( c \)
- The distance from the center \( (2, 0) \) to the vertex \( (1, 0) \) is \( a \):
\[
a = |2 - 1| = 1
\]
- The distance from the center \( (2, 0) \) to the focus \( (3, 0) \) is \( c \):
\[
c = |2 - 3| = 1
\]

### Step 4: Find \( b \) Using the Relationship
Using the relationship \( c^2 = a^2 + b^2 \):
\[
1^2 = 1^2 + b^2
\]
\[
1 = 1 + b^2
\]
\[
b^2 = 0
\]
Since \( b^2 = 0 \), this indicates that the hyperbola degenerates into a pair of intersecting lines, which is a special case. However, the provided data suggests a scenario where the hyperbola is very narrow.

### Step 5: Form the Equation
Given that \( b = 0 \), the equation is not a typical hyperbola but rather can be represented as:
\[
\frac{(x - 2)^2}{1} - \frac{(y - 0)^2}{0} = 1
\]
This simplifies to:
\[
(x - 2)^2 = 1
\]
Or:
\[
x = 2 \pm 1
\]
Which are vertical lines at \( x = 3 \) and \( x = 1 \).

### Conclusion
The nature of the data given suggests a unique situation, but based on the usual approach, the equation of the hyperbola or pair of lines considering this scenario is \( (x - 2)^2 = 1 \), representing lines \( x = 3 \) and \( x = 1 \).

Would you like to explore this concept further or need any additional details?

### Follow-up Questions:
1. How does the hyperbola differ from an ellipse?
2. What are the asymptotes of a hyperbola, and how are they found?
3. How would the equation change if the hyperbola was vertical?
4. Can you derive the equation of a hyperbola given only its asymptotes?
5. What happens when the distance between the foci increases?

### Tip:
For hyperbolas, always double-check the relationship \( c^2 = a^2 + b^2 \) to ensure the values are consistent with the given parameters.

Explore how to determine the equation of a hyperbola with (3,0) as a focus and (1,0) as a vertex. Learn about the structure of hyperbolas, center calculation, and the relationship between focus, vertex, and center. Suitable for advanced high school students.

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