Distance between foci of a hyperbola is double the distance between its vertices . If the length of its conjugate axis is 6, then equation is

Options

A. `3x^(2) - y^(2) = 3`
B. `x^(2) - 3y^(2) = 3`
C. `3x^(2) - y^(2) = 9`
D. `x^(2) - 3y^(2) = 9`

Answer

To find the equation of the hyperbola given the conditions in the problem, we will follow these steps:

### Step 1: Understand the properties of the hyperbola
The standard form of the equation of a hyperbola that opens horizontally is given by:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
where \(2a\) is the distance between the vertices and \(2c\) is the distance between the foci.

### Step 2: Relate the distances
We are given that the distance between the foci is double the distance between its vertices. Therefore, we can write:
\[ 2c = 2 \times 2a \implies c = 2a \]

### Step 3: Use the length of the conjugate axis
The length of the conjugate axis is given as 6. The length of the conjugate axis is defined as \(2b\), so:
\[ 2b = 6 \implies b = 3 \]

### Step 4: Use the relationship between \(a\), \(b\), and \(c\)
For hyperbolas, the relationship between \(a\), \(b\), and \(c\) is given by:
\[ c^2 = a^2 + b^2 \]
Substituting \(c = 2a\) and \(b = 3\) into this equation gives:
\[ (2a)^2 = a^2 + 3^2 \]
\[ 4a^2 = a^2 + 9 \]

### Step 5: Solve for \(a^2\)
Rearranging the equation:
\[ 4a^2 - a^2 = 9 \implies 3a^2 = 9 \implies a^2 = 3 \]

### Step 6: Write the equation of the hyperbola
Now that we have \(a^2\) and \(b^2\), we can substitute these values into the standard form of the hyperbola:
\[ \frac{x^2}{3} - \frac{y^2}{9} = 1 \]
To express this in a more standard form, we can multiply through by 9:
\[ 3x^2 - y^2 = 9 \]

### Final Answer
Thus, the equation of the hyperbola is:
\[ 3x^2 - y^2 = 9 \]