The vertices of a hyperbola are at (0, 0) and (10, 0) and one of its foci is at (18, 0). The equation of the hyperbola is

Options

A. `(x^2)/(25)-(y^2)/(144)=1`
B. `((x-5)^2)/(25)-(y^2)/(144)=1`
C. `(x^2)/(5)-((y-5)^2)/(144)=1`
D. `((x-5)^2)/(25)-((y-5)^2)/(144)=1`

Answer

To find the equation of the hyperbola given the vertices and one of the foci, we can follow these steps:

### Step 1: Identify the center of the hyperbola
The vertices of the hyperbola are given at (0, 0) and (10, 0). The center of the hyperbola is the midpoint of the line segment joining the vertices.

- Midpoint (center) = \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)
- Here, \(x_1 = 0\), \(y_1 = 0\), \(x_2 = 10\), \(y_2 = 0\)

Calculating the midpoint:
\[ \text{Center} = \left(\frac{0 + 10}{2}, \frac{0 + 0}{2}\right) = (5, 0) \]

### Step 2: Determine the distance \(a\)
The distance from the center to each vertex is \(a\). Since the vertices are at (0, 0) and (10, 0), the distance from the center (5, 0) to either vertex is:
\[ a = 5 \]

### Step 3: Identify the distance \(c\) to the focus
The distance from the center to a focus is denoted as \(c\). One of the foci is given at (18, 0). The distance from the center (5, 0) to the focus (18, 0) is:
\[ c = 18 - 5 = 13 \]

### Step 4: Calculate \(b\) using the relationship \(c^2 = a^2 + b^2\)
We know the relationship between \(a\), \(b\), and \(c\) for hyperbolas:
\[ c^2 = a^2 + b^2 \]
Substituting the known values:
\[ 13^2 = 5^2 + b^2 \]
\[ 169 = 25 + b^2 \]
\[ b^2 = 169 - 25 = 144 \]

### Step 5: Write the equation of the hyperbola
The standard form of the equation of a hyperbola centered at \((h, k)\) with a horizontal transverse axis is:
\[ \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 \]
Substituting \(h = 5\), \(k = 0\), \(a^2 = 25\), and \(b^2 = 144\):
\[ \frac{(x - 5)^2}{25} - \frac{y^2}{144} = 1 \]

### Final Answer
The equation of the hyperbola is:
\[ \frac{(x - 5)^2}{25} - \frac{y^2}{144} = 1 \]