The equation of the hyperola whose centre is (1,2) one focus is (6,2) and transverse axis 6 is

Options

A. `16(x-1)^(2)-9(y-2)^(2)=144`
B. `9(x-1)^(2)-16(y-2)^(2)=144`
C. `16(x-1)^(2)-25(y-2)^(2)=200`
D. `25(x-1)^(2)-16(y-2)^(2)=200`

Answer

To find the equation of the hyperbola with the given parameters, we will follow these steps:

### Step 1: Identify the given information
- Center of the hyperbola: \( (1, 2) \)
- Focus of the hyperbola: \( (6, 2) \)
- Length of the transverse axis: \( 6 \)

### Step 2: Determine the value of \( A \)
The length of the transverse axis is given by \( 2A \). Since the transverse axis is \( 6 \):
\[ 2A = 6 \implies A = \frac{6}{2} = 3 \]

### Step 3: Calculate the distance from the center to the focus
The distance from the center to the focus is denoted as \( c \). The center is \( (1, 2) \) and the focus is \( (6, 2) \). The distance \( c \) can be calculated as follows:
\[ c = \text{distance between the center and the focus} = 6 - 1 = 5 \]

### Step 4: Use the relationship between \( A \), \( B \), and \( C \)
For hyperbolas, the relationship between \( A \), \( B \), and \( C \) is given by:
\[ c^2 = a^2 + b^2 \]
where \( c = 5 \) and \( A = 3 \) (so \( A^2 = 3^2 = 9 \)):
\[ 5^2 = 3^2 + B^2 \implies 25 = 9 + B^2 \implies B^2 = 25 - 9 = 16 \implies B = 4 \]

### Step 5: Write the standard form of the hyperbola
Since the transverse axis is horizontal (because the foci have the same y-coordinate), the standard form of the hyperbola is:
\[ \frac{(x - h)^2}{A^2} - \frac{(y - k)^2}{B^2} = 1 \]
where \( (h, k) \) is the center. Substituting \( h = 1 \), \( k = 2 \), \( A = 3 \), and \( B = 4 \):
\[ \frac{(x - 1)^2}{3^2} - \frac{(y - 2)^2}{4^2} = 1 \]
This simplifies to:
\[ \frac{(x - 1)^2}{9} - \frac{(y - 2)^2}{16} = 1 \]

### Step 6: Rearranging to standard form
To express this in a more standard form, we can multiply through by 144 (the least common multiple of 9 and 16):
\[ 16(x - 1)^2 - 9(y - 2)^2 = 144 \]

### Final Equation
Thus, the equation of the hyperbola is:
\[ 16(x - 1)^2 - 9(y - 2)^2 = 144 \]