The directrix of the hyperbola `(x ^(2))/(9) - (y ^(2))/(4) =1` is

Options

A. `x = 9sqrt13`
B. `y = (9)/(sqrt13)`
C. `x= 6sqrt13`
D. `y = (6)/(sqrt13)`

Answer

To find the directrix of the hyperbola given by the equation

\[ \frac{x^2}{9} - \frac{y^2}{4} = 1, \]

we follow these steps:

### Step 1: Identify the values of \(a\) and \(b\)

The standard form of a hyperbola is

\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1. \]

From the given equation, we can identify:

\[ a^2 = 9 \quad \text{and} \quad b^2 = 4. \]

Thus, we find:

\[ a = \sqrt{9} = 3 \quad \text{and} \quad b = \sqrt{4} = 2. \]

### Step 2: Calculate the eccentricity \(e\)

The eccentricity \(e\) of a hyperbola is given by the formula:

\[ e = \sqrt{1 + \frac{b^2}{a^2}}. \]

Substituting the values of \(b^2\) and \(a^2\):

\[ e = \sqrt{1 + \frac{4}{9}} = \sqrt{1 + \frac{4}{9}} = \sqrt{\frac{9 + 4}{9}} = \sqrt{\frac{13}{9}} = \frac{\sqrt{13}}{3}. \]

### Step 3: Write the equations of the directrices

The equations of the directrices for the hyperbola are given by:

\[ x = \pm \frac{a}{e}. \]

Substituting the values of \(a\) and \(e\):

\[ x = \pm \frac{3}{\frac{\sqrt{13}}{3}} = \pm \frac{3 \cdot 3}{\sqrt{13}} = \pm \frac{9}{\sqrt{13}}. \]

### Step 4: Final answer

Thus, the equations of the directrices of the hyperbola are:

\[ x = \frac{9}{\sqrt{13}} \quad \text{and} \quad x = -\frac{9}{\sqrt{13}}. \]